Method, device and system for trajectory planning of a robot arm, and storage medium

Abstract

This application discloses a method, apparatus, system, and storage medium for planning the motion trajectory of a robotic arm, relating to the field of mechanical control technology. The method includes acquiring an initial path point, an end path point, and intermediate path points; calculating the absolute distance from each path point to the initial path point and the total absolute distance; setting a preset spatial velocity; calculating the total travel time and the time to reach each path point; setting a preset speed change time; calculating the required speed change time for each path point; calculating the angle vector of the axis motor in the corresponding joint space and the velocity and acceleration of the axis motor in the uniform and variable speed segments of each path segment; setting a preset sampling time; calculating the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point; and planning the motion trajectory of the robotic arm. The embodiments of this application improve the accuracy and efficiency of robotic arm movement, reduce vibration and impact, and improve the stability and safety of movement.

Description

Technical Field

[0001] This application belongs to the field of mechanical control technology, and in particular relates to a method, device, system and storage medium for planning the motion trajectory of a robotic arm. Background Technology

[0002] With the widespread application of low-altitude production services supported by drones in industries and agriculture, robotic arms, as an important payload for drones, have received extensive research and attention, further expanding the application scenarios of drones.

[0003] Traditional robotic arm trajectory planning uses trinomial polynomial interpolation, quintic polynomial interpolation, and Cartesian trajectory planning. It requires initial values ​​for the position, velocity, and acceleration of each path point, resulting in numerous parameters, excessive calculation time, and the tendency to jitter and repeat during movement. This affects the lifespan of the joint motors and interferes with the UAV control system, impacting the flight stability of the UAV and potentially leading to crashes.

[0004] Therefore, given the development of the drone industry and the shortcomings of traditional trajectory planning algorithms for robotic arms, there is an urgent need for an optimized algorithm to control drone robotic arms. Summary of the Invention

[0005] The purpose of this application is to provide a method, device, system and storage medium for planning the motion trajectory of a robotic arm. By giving spatial velocity and speed change time, the trajectory planning of the UAV robotic arm is performed, reducing parameter assignment and realizing the synchronization of different axis joints in the joint space, ensuring continuity at each path point, and making the position and velocity motion trajectory of the robotic arm smooth and with little jitter during operation, thus meeting the application requirements of UAV robotic arms.

[0006] To achieve the above objectives, the solution proposed in this application is:

[0007] In a first aspect, embodiments of this application provide a method for planning the motion trajectory of a robotic arm, including:

[0008] Obtain the positions of n path points, including the initial path point, the final path point, and n-2 intermediate path points; calculate the absolute distance from each path point to the initial path point and the total absolute distance based on the n path point positions;

[0009] Set a preset spatial speed and calculate the total time spent based on the preset spatial speed and the total absolute distance; calculate the time to reach each path point based on the absolute distance from each path point to the initial path point, the total absolute distance, and the total time spent; set a preset speed change time and calculate the required speed change time for each path point based on the preset speed change time.

[0010] Calculate the angle vectors of the m-axis motors in the joint space corresponding to the n path points of the robotic arm in Cartesian space, and the velocities and accelerations of the m-axis motors in the uniform and variable speed segments of each path segment.

[0011] The system presets a sampling time and calculates the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point. Based on these values, the system plans the motion trajectory of the robotic arm.

[0012] The method described in the embodiments of this application may also have the following additional technical features:

[0013] Furthermore, based on the positions of the n path points, calculate the absolute distance from each path point to the initial path point and the total absolute distance, including:

[0014] Let p be the coordinates of n path points in Cartesian space. i (x i y i , z i Let i = 1, 2, ..., n, then the formula for the absolute distance from each path point to the initial path point is as follows:

[0015]

[0016] Total absolute distance d total =d n The formula for the distance vector is as follows:

[0017] d = [d1 d2 ... d] n ]

[0018] Where, d i d represents the absolute distance from the initial path point to path point i. total The distance represents the total absolute distance from the initial path point to the final path point, and d represents the distance vector.

[0019] Furthermore, the preset space velocity is set to S. target The formula for the total time spent is as follows:

[0020] t total =d total / S target

[0021] Among them, t total S represents the total time spent. target Indicates the preset space velocity;

[0022] The formulas for the arrival time at each waypoint are shown below:

[0023]

[0024] Among them, t i Indicates the time to reach path point i;

[0025] The formula for the time vector is shown below:

[0026] t = [t1 t2 ... t] n ]

[0027] Where t represents the time vector.

[0028] Furthermore, a preset speed change time is set, and the required speed change time for each path point is calculated based on the preset speed change time, including:

[0029] Set the preset speed change time to T acc The formula for the required change-of-speed time at each path point is as follows:

[0030]

[0031] Among them, T acc This represents the preset speed change time; when i = 1, t acci This represents the acceleration time at the initial path point; when i = n, t acci This represents the deceleration time at the end path point; when 2 ≤ i ≤ n-1, t acci This indicates the time of acceleration / deceleration at intermediate path points;

[0032] The formula for the variable-speed time vector is shown below:

[0033] t acc =[t acc1 t acc2 ... t accn ]

[0034] Among them, t acc This represents the variable speed time vector.

[0035] Furthermore, before calculating the required shift time for each path point based on the preset shift time, a time constraint condition is determined. The time constraint condition includes that the shift time between two path points is less than the total time spent between the two path points, as shown in the following formula:

[0036]

[0037] After satisfying the time constraints, calculate the velocity of the uniform segment and the acceleration of the variable segment for each path segment.

[0038] Furthermore, the calculation includes the angle vectors of the m-axis motors in the joint space corresponding to the n path points in Cartesian space, and the velocities and accelerations of the m-axis motors in the uniform and variable speed segments of each path segment, including:

[0039] The angle vector θ of the m-axis motors in the joint space corresponding to the n path points in Cartesian space of the robotic arm is obtained by inverse kinematics. i The formula is shown below:

[0040] θ i =[θ i1 θ i2 ... θ im ]

[0041] Where i = 1, 2, ..., n;

[0042] The path points are connected by n-1 path segments. The speed formulas for the constant speed segments of the m-axis motors in each path segment are shown below:

[0043] v j =[v j1 v j2 ... v jm ]=(θ j+1 -θ j ) / (t j+1 -t j )

[0044] Where j = 1, 2, ..., n-1;

[0045] The speed change segments of the m-axis motors in each path segment include the acceleration segment from the initial path point to the intermediate path point, the deceleration segment from the intermediate path point to the end path point, and the speed change segment between the first intermediate path point and the last intermediate path point.

[0046] The acceleration vector formula for an m-axis motor during the acceleration segment from the initial path point to the intermediate path point is shown below:

[0047] a start =[a start1 a start2 ... a startm ] = v1 / t acc1 ;

[0048] The formula for the acceleration vector of a motor with m axes during the deceleration segment from the intermediate path point to the final path point is shown below:

[0049] a end =[a end1 a end2 ... a endm ]=(0-v n-1 ) / taccn ;

[0050] The acceleration vector formula for a motor with m axes during the speed change segment between the first intermediate path point and the last intermediate path point is shown below:

[0051] a middle =[a middle1 a middle2 ... a middlem ]=(v k -v k-1 ) / t acck ;

[0052] Where k = 2, 3, ..., n-1.

[0053] Furthermore, a preset sampling time is used to calculate the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point, including:

[0054] Set the preset sampling time to t;

[0055] When the robotic arm is in the acceleration phase from the initial path point to the first intermediate path point and t < t acc1 The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0056] a real =[a real1 a real2 ... a realm ] = a start

[0057] The formula for the real-time value of the velocity vector is shown below:

[0058] v real =[v real1 v real2 ... v realm ] = a start ·t

[0059] The formula for the real-time value of the position vector is shown below:

[0060]

[0061] When the robotic arm moves at a constant speed in the segment from the initial path point to the first intermediate path point, and t acc1 ≤t<(t2-t acc2 When / 2), the formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0062] a real =0

[0063] The formula for the real-time value of the velocity vector is shown below:

[0064] v real =v1

[0065] The formula for the real-time value of the position vector is shown below:

[0066] θ real =θ1+v1·t

[0067] When the robotic arm moves in the variable speed segment between the two midpoints and (t) i -t acci / 2)≤t<(t i +t acci When i = 2, 3, ..., n-1, the formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0068] a real =a middle

[0069] The formula for the real-time value of the velocity vector is shown below:

[0070] v real =v i-1 +a middle ·(tt i +t acci / 2)

[0071] The formula for the real-time value of the position vector is shown below:

[0072]

[0073] When the robotic arm moves at a constant speed between the two midpoints and (t) i +t acci / 2)≤t<(t i+1 -t acc(i+1) When i = 2, 3, ..., n-1, the formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0074] a real =0

[0075] The formula for the real-time value of the velocity vector is shown below:

[0076] v real =v n-1

[0077] The formula for the real-time value of the position vector is shown below:

[0078] θ real =θ n-1 +v n-1 ·(tt n-1 )

[0079] When the robotic arm is in the deceleration phase from the last intermediate path point to the end path point and (t) n -t accn )≤t≤t n The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0080] a real =a end

[0081] The formula for the real-time value of the velocity vector is shown below:

[0082] v real =v n-1 +a end ·(tt n +t accn )

[0083] The formula for the real-time value of the position vector is shown below:

[0084]

[0085] Secondly, embodiments of this application provide a motion trajectory planning device for a robotic arm, comprising:

[0086] The distance calculation module is configured to obtain the positions of n path points, including the initial path point, the final path point, and n-2 intermediate path points; and to calculate the absolute distance from each path point to the initial path point and the total absolute distance based on the n path point positions.

[0087] The time calculation module is configured to set a preset spatial speed, calculate the total time spent based on the preset spatial speed and the total absolute distance; calculate the time to reach each path point based on the absolute distance from each path point to the initial path point, the total absolute distance, and the total time spent; and set a preset speed change time, calculate the required speed change time for each path point based on the preset speed change time.

[0088] The speed calculation module is configured to calculate the angle vectors of the m-axis motors in the joint space corresponding to the n path points in the Cartesian space of the robotic arm, as well as the constant speed segment velocity and variable speed segment acceleration of the m-axis motors in each path segment.

[0089] The trajectory planning module is configured to use a preset sampling time to calculate the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point, and to plan the motion trajectory of the robotic arm based on these values.

[0090] Thirdly, embodiments of this application provide a motion trajectory planning system for a robotic arm. The system includes a processor and a memory. The memory stores a computer program, which is loaded and executed by the processor to implement the motion trajectory planning method for the robotic arm as provided in the first aspect of embodiments of this application.

[0091] Fourthly, embodiments of this application provide a computer-readable storage medium storing a computer program, which, when executed by a processor, is used to implement the motion trajectory planning method for a robotic arm as described in the first aspect of embodiments of this application.

[0092] The motion trajectory planning method for the robotic arm provided in this application has the following advantages compared with the prior art:

[0093] This application embodiment obtains n precise path point positions, including the initial path point, the end path point, and n-2 intermediate path points, which ensures that the robotic arm accurately passes through these points during movement, achieving high-precision task execution; by converting the path points in Cartesian space into parameters in joint space, and obtaining precise joint angle vectors through inverse kinematics calculation, the accuracy of the robotic arm's movement is further improved.

[0094] This embodiment of the application optimizes the movement time of the robotic arm and improves work efficiency by setting a preset spatial speed, calculating the total time spent based on the total absolute distance, and allocating arrival time according to the absolute distance of each path point. By setting a preset speed change time, the robotic arm can smoothly decelerate before reaching each path point and accelerate when leaving, reducing unnecessary energy consumption and time waste.

[0095] This application embodiment calculates the joint acceleration vector, velocity vector, and position vector at each time point in real time by preset sampling time, which can ensure that the robotic arm maintains a smooth trajectory during movement and reduce shaking and impact. By calculating the constant speed segment velocity and variable speed segment acceleration of each path segment, the movement of the robotic arm can be further smoothed, improving the stability and safety of the movement.

[0096] This application supports any number of path points, including initial path points, end path points, and intermediate path points, enabling the robotic arm to flexibly cope with various complex motion requirements; wherein preset spatial speed, speed change time, and other parameters can be adjusted according to actual needs to adapt to different working environments and task requirements.

[0097] The embodiments of this application can effectively avoid collisions with the surrounding environment, improve work safety, and ensure the stable operation of the robotic arm by precisely controlling the movement trajectory and speed of the robotic arm. Attached Figure Description

[0098] Figure 1 A flowchart illustrating the motion trajectory planning method for a robotic arm according to an embodiment of this application is shown;

[0099] Figure 2 A structural block diagram of the motion trajectory planning device for a robotic arm according to an embodiment of this application is shown;

[0100] Figure 3 A structural block diagram of a computer device according to an embodiment of this application is shown;

[0101] Figure 4 A joint position graph of an embodiment of this application is shown;

[0102] Figure 5 The joint position curves of a quintic polynomial under the same conditions are shown.

[0103] Figure 6 A joint velocity curve diagram of an embodiment of this application is shown;

[0104] Figure 7 The joint velocity curves of a fifth-order polynomial under the same conditions are shown.

[0105] Figure 8 The diagram shows the joint acceleration curves of an embodiment of this application;

[0106] Figure 9 The joint acceleration curves of a fifth-order polynomial under the same conditions are shown. Detailed Implementation

[0107] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the specific embodiments of this application will be described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of this application. Furthermore, it should be noted that, for ease of description, only the parts relevant to this application are shown in the accompanying drawings, not the entire structure. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without inventive effort are within the scope of protection of this application.

[0108] The terms “comprising” and “having”, and any variations thereof, used in this application are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or apparatus that includes a series of steps or units is not limited to the steps or units listed, but may optionally include steps or units not listed, or may optionally include other steps or units inherent to such process, method, product, or apparatus.

[0109] In this application, the reference to "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a mutually exclusive, independent, or alternative embodiment. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described in this application can be combined with other embodiments.

[0110] like Figure 1 As shown in the figure, this application provides a method for planning the motion trajectory of a robotic arm, including the following steps:

[0111] Step 101: Obtain the positions of n path points, including the initial path point, the final path point, and n-2 intermediate path points; calculate the absolute distance from each path point to the initial path point and the total absolute distance based on the n path point positions.

[0112] In path planning for robots or robotic arms, acquiring and processing the locations of waypoints is a crucial step. This process involves determining a series of points (often called waypoints) that define the trajectory that the robotic arm or robot needs to follow.

[0113] First, the specific locations of the n path points need to be determined. These locations are typically defined in three-dimensional space, meaning each path point has x, y, and z coordinates. These path points can be obtained in various ways, including but not limited to:

[0114] Manual input: Users can directly input the coordinates of each path point in the control software.

[0115] Sensor data: If a robot or robotic arm is equipped with appropriate sensors (such as vision sensors, lidar, etc.), it can automatically identify and locate waypoints by sensing the surrounding environment.

[0116] Path planning algorithms: In some cases, path planning algorithms (such as A* algorithm, RRT algorithm, etc.) can be used to automatically generate a series of waypoints to meet specific task requirements.

[0117] Once the positions of all path points are obtained, the next step is to calculate the absolute distance from each path point to the initial path point. Absolute distance refers to the straight-line distance between two points, which can be calculated using the Euclidean distance formula in three-dimensional space.

[0118] The total absolute distance is the sum of the absolute distances from all path points to the initial path point. This value can be used to assess the length of the entire path or as a basis for calculating the time required for a robotic arm or robot to move.

[0119] In this embodiment, the position coordinates of n path points in Cartesian space are set as p. i (x i y i , z i Let i = 1, 2, ..., n, then the formula for the absolute distance from each path point to the initial path point is as follows:

[0120]

[0121] Total absolute distance d total =d n The formula for the distance vector is as follows:

[0122] d = [d1 d2 ... d] n ]

[0123] Where, d i d represents the absolute distance from the initial path point to path point i. total The distance represents the total absolute distance from the initial path point to the final path point, and d represents the distance vector.

[0124] Specifically, this application uses the three path points of a six-axis robotic arm in Cartesian space as an example for illustration.

[0125] Let the three path points be the initial path point p1(-0.6, -0.2, 0), the intermediate path point p2(-0.6, -0.4, -0.4), and the final path point p3(0.5, -0.4, 0.1). Then the absolute distances from each path point to the initial path point are: d1 = 0, d2 = 0.4472, d3 = 1.6555, d total =d3=1.6555, distance vector d=[0 0.4472 1.6555].

[0126] Step 102: Set a preset spatial speed, calculate the total time spent based on the preset spatial speed and the total absolute distance; calculate the time to reach each path point based on the absolute distance from each path point to the initial path point, the total absolute distance, and the total time spent; set a preset speed change time, calculate the speed change time required for each path point based on the preset speed change time.

[0127] In the motion trajectory planning of robotic arms or robots, setting a preset spatial speed, calculating the total time spent, allocating the time to reach each path point, and setting a preset speed change time and calculating the speed change time are key steps to ensure that the robotic arm can move smoothly and efficiently along the predetermined trajectory.

[0128] The preset spatial speed refers to the distance the robotic arm moves in space per second during its uniform motion phase. The selection of this speed needs to consider factors such as the robotic arm's performance limitations (e.g., maximum speed, acceleration limits), task requirements (e.g., accuracy requirements, time constraints), and the working environment (e.g., obstacle distribution, space constraints). The preset spatial speed is typically set in meters per second (m / s).

[0129] Total time refers to the total time required for the robotic arm to move from the initial path point to the final path point. This is based on the preset spatial velocity and the total absolute distance (i.e., the sum of the straight-line distances between all path points).

[0130] After determining the total time, the time to reach each waypoint needs to be allocated based on the absolute distance from each waypoint to the initial waypoint. This is typically achieved by multiplying the ratio of the absolute distance of each waypoint to the total absolute distance by the total time.

[0131] The preset speed change time refers to the time required for the robotic arm to decelerate before reaching each waypoint and accelerate upon leaving. This time setting needs to take into account the robotic arm's acceleration limitations and the smoothness requirements of the speed change process. The preset speed change time is typically set in seconds (s).

[0132] Although the speed change time may vary in practical applications due to the dynamic characteristics and control strategy of the robotic arm, it can generally be assumed that a fixed speed change time is allocated before and after each waypoint. This means that before each waypoint, the robotic arm will begin to decelerate and reduce its speed to zero (or close to zero) when it reaches that waypoint; then, when leaving that waypoint, the robotic arm will begin to accelerate until it reaches the preset spatial velocity.

[0133] In this embodiment, the preset spatial velocity is set to S. target The formula for the total time spent is as follows:

[0134] t total =d total / S target

[0135] Among them, t total S represents the total time spent. target Indicates the preset space velocity;

[0136] The formulas for the arrival time at each waypoint are shown below:

[0137]

[0138] Among them, t i Indicates the time to reach path point i;

[0139] The formula for the time vector is shown below:

[0140] t = [t1 t2 ... t] n ]

[0141] Where t represents the time vector.

[0142] Set the preset speed change time to T acc The formula for the required change-of-speed time at each path point is as follows:

[0143]

[0144] Among them, T acc This represents the preset speed change time; when i = 1, t acci This represents the acceleration time at the initial path point; when i = n, t acci This represents the deceleration time at the end path point; when 2 ≤ i ≤ n-1, t acci This indicates the time of acceleration / deceleration at intermediate path points;

[0145] The formula for the variable-speed time vector is shown below:

[0146] t acc =[t acc1 t acc2 ... t accn ]

[0147] Among them, t acc This represents the variable speed time vector.

[0148] Specifically, the preset spatial velocity S target =0.1, preset shift time is T acc =0.8, then the total time t total =d total / S target =1.6555 / 0.1=16.555, the arrival times at each path point are t1=0, t2=4, t3=16.555, t=[0 4 16.555], and the required change-of-speed time t at each path point is... acc1 =0.4, t acc2 =0.8, t acc3 =0.4, t acc = [0.4 0.8 0.4], where t acc1 Including only the acceleration time in the initial phase, t acc3 Only the deceleration time at the end of the phase is included, t acc2 The time for speed change (both acceleration and deceleration are possible) between the constant speed segments of the two paths.

[0149] Before calculating the required shift time for each path point based on the preset shift time, this embodiment of the application also performs a time constraint judgment. The time constraint condition includes that the shift time between two path points is less than the total time spent between the two path points, as shown in the following formula:

[0150]

[0151] After satisfying the time constraints, calculate the velocity of the uniform segment and the acceleration of the variable segment for each path segment.

[0152] The formula for the time constraint in this embodiment is as follows:

[0153]

[0154] The time constraint expressed in the above formula is valid.

[0155] Step 103: Calculate the angle vectors of the m axis motors in the joint space corresponding to the n path points in the Cartesian space of the robotic arm, and the velocity and acceleration of the m axis motors in the uniform and variable speed segments of each path segment.

[0156] In the motion trajectory planning of a robotic arm, converting path points in Cartesian space into parameters in joint space is a crucial step. Joint space refers to the space formed by the angles or positions of the joints of the robotic arm, which directly determines the position and orientation of the end effector (such as a gripper or welding torch) in Cartesian space.

[0157] For a robotic arm with m joints, each joint has a corresponding angle or position parameter. When the robotic arm moves through n path points in Cartesian space, we need to calculate the angle vector in joint space for each path point. This angle vector is an array containing m elements, where each element represents the angle value of a joint.

[0158] The calculation of these angle values ​​typically involves solving inverse kinematics (IK). Inverse kinematics is an important concept in robotics, referring to the deduction of the angles or positions of the joints of a robotic arm from the position and orientation of its end effector in Cartesian space. Solving inverse kinematics can involve complex mathematical operations and algorithms, such as numerical methods and analytical methods.

[0159] During the movement of a robotic arm, there are usually constant speed and variable speed segments. The constant speed segment refers to the period when the robotic arm moves at a constant speed, while the variable speed segment refers to the period when the speed of the robotic arm changes, including acceleration and deceleration.

[0160] Constant speed segment: During the constant speed segment, the speed of each axis motor of the robotic arm remains constant. This speed value can be set according to factors such as task requirements, robotic arm performance, and working environment. In joint space, this speed value will be converted into the rate of change of angular velocity of each joint.

[0161] Acceleration during speed changes: In speed change segments, the robotic arm needs to accelerate or decelerate to reach or leave the speed range of the constant speed segment. This acceleration or deceleration process needs to be smooth and controllable to avoid excessive impact or vibration on the robotic arm. Therefore, it is necessary to calculate the acceleration during speed change segments to ensure that the robotic arm can smoothly transition to the next speed stage. Acceleration calculations are typically based on the robotic arm's dynamic characteristics and control strategies.

[0162] This application embodiment obtains the angle vector θ of the m-axis motors in the joint space corresponding to the n path points in Cartesian space of the robotic arm through inverse kinematics. i The formula is shown below:

[0163] θ i =[θ i1 θ i2 ... θ im ]

[0164] Where i = 1, 2, ..., n;

[0165] The path points are connected by n-1 path segments. The speed formulas for the constant speed segments of the m-axis motors in each path segment are shown below:

[0166] v j =[v j1 v j2 ... v jm ]=(θ j+1 -θ j ) / (t j+1 -t j )

[0167] Where j = 1, 2, ..., n-1;

[0168] The speed change segments of the m-axis motors in each path segment include the acceleration segment from the initial path point to the intermediate path point, the deceleration segment from the intermediate path point to the end path point, and the speed change segment between the first intermediate path point and the last intermediate path point.

[0169] The acceleration vector formula for an m-axis motor during the acceleration segment from the initial path point to the intermediate path point is shown below:

[0170] a start =[a start1 a start2 ... a startm ] = v1 / t acc1 ;

[0171] The formula for the acceleration vector of a motor with m axes during the deceleration segment from the intermediate path point to the final path point is shown below:

[0172] a end =[a end1 a end2 ... a endm ]=(0-v n-1 ) / t accn ;

[0173] The acceleration vector formula for a motor with m axes during the speed change segment between the first intermediate path point and the last intermediate path point is shown below:

[0174] a middle =[a middle1 a middle2 ... a middlem ]=(v k -v k-1 ) / t acck ;

[0175] Where k = 2, 3, ..., n-1.

[0176] Specifically, through inverse kinematics, the angle vectors of the six axis motors in the joint space corresponding to the initial path point p1 (-0.6, -0.2, 0), intermediate path point p2 (-0.6, -0.4, -0.4), and end path point p3 (0.5, -0.4, 0.1) of the six-axis robotic arm in Cartesian space are obtained as follows:

[0177] θ1=[θ 11 θ 12 θ 13 θ 14 θ 15 θ 16 = [-3.06 -0.78 -0.05 3.14 2.30 3.06]

[0178] θ2=[θ 21 θ 22 θ 23 θ 24 θ 25 θ 26 = [-2.76 0.16 -0.90 3.14 2.40 2.84]

[0179] θ3=[θ 31 θ 32 θ 33 θ 34 θ 35 θ 36= [-3.58 2.54 -0.11 3.52 -1.70 3.42]

[0180] Where, θ 12 θ represents the rotation angle corresponding to the second axis joint at path point 1. 13 This indicates the rotation angle corresponding to the third axis joint at path point 1, and so on.

[0181] There are two path segments between the three path points, namely path segment a and path segment b. The speed formulas for the six-axis motors in the constant speed segment of each path segment are shown below:

[0182] v a =[v a1 v a2 v a3 v a4 v a5 v a6 = [0.07 0.24 -0.21 0 0.02 -0.06]

[0183] v b =[v b1 v b2 v b3 v b4 v b5 v b6 = [-0.07 0.19 0.06 0.03 -0.33 0.05]

[0184] Among them, v a1 This represents the velocity v of joint 1 in the uniform segment corresponding to path segment a. b2 This indicates the velocity of joint 2 in the uniform segment corresponding to path segment b, and so on.

[0185] The speed-changing section consists of three parts: the initial acceleration phase, the speed-changing section between the two constant-speed phases, and the final deceleration phase. The acceleration formulas for the six-axis motors in each speed-changing section are shown below: a start =[a start1 a start2 a start3 a start4 a start5 a start6 ]

[0186] =[0.19 0.59 -0.53 0 0.06 -0.14]

[0187] a middle =[a middle1 a middle2 a middle3 a middle4 amiddle5 a middle6 ]

[0188] = [-0.17 -0.06 0.34 0.04 -0.44 0.13]

[0189] a end =[a end1 a end2 a end3 a end4 a end5 a end6 ]

[0190] =[0.16 -0.48 -0.16 -0.08 0.82 -0.12]

[0191] Among them, a start a represents the acceleration vector during the initial acceleration phase. middle a represents the acceleration vector during the intermediate speed change segment. end This represents the acceleration vector at the end of the acceleration phase.

[0192] Step 104: Preset sampling time, calculate the real-time values ​​of joint acceleration vector, velocity vector and position vector of the robotic arm at each time point according to the preset sampling time, and plan the motion trajectory of the robotic arm according to the real-time values ​​of joint acceleration vector, velocity vector and position vector of the robotic arm at each time point.

[0193] In the motion control of robotic arms, to ensure that the robotic arm can move smoothly and accurately along a predetermined trajectory, a time-sampling-based method is used to calculate and adjust the joint parameters of the robotic arm in real time. This process involves setting a preset sampling time, calculating the real-time values ​​of joint acceleration vectors, velocity vectors, and position vectors, and planning the motion trajectory of the robotic arm based on these values.

[0194] The preset sampling time is a fixed time interval used to periodically collect data and update control parameters during the movement of the robotic arm. The choice of this time interval depends on several factors, including the dynamic performance of the robotic arm, the complexity of the control algorithm, and the accuracy and real-time requirements of the task.

[0195] At each sampling time point, the real-time values ​​of the acceleration vector, velocity vector, and position vector of each joint of the robotic arm are calculated based on the current state of the robotic arm (such as joint angle, angular velocity, angular acceleration, etc.). Based on the calculated real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point, the motion trajectory of the robotic arm is planned.

[0196] In this embodiment, the preset sampling time is set to t;

[0197] When the robotic arm is in the acceleration phase from the initial path point to the first intermediate path point and t < t acc1 The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0198] a real =[a real1 a real2 ... a realm ] = a start

[0199] The formula for the real-time value of the velocity vector is shown below:

[0200] v real =[v real1 v real2 ... v realm ] = a start ·t

[0201] The formula for the real-time value of the position vector is shown below:

[0202]

[0203] When the robotic arm moves at a constant speed in the segment from the initial path point to the first intermediate path point, and t acc1 ≤t<(t2-t acc2 When / 2), the formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0204] a real =0

[0205] The formula for the real-time value of the velocity vector is shown below:

[0206] v real =v1

[0207] The formula for the real-time value of the position vector is shown below:

[0208] θ real =θ1+v1·t

[0209] When the robotic arm moves in the variable speed segment between the two midpoints and (t) i -t acci / 2)≤t<(t i +t acci When i = 2, 3, ..., n-1, the formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0210] a real =a middle

[0211] The formula for the real-time value of the velocity vector is shown below:

[0212] v real =v i-1 +a middle ·(tt i +t acci / 2)

[0213] The formula for the real-time value of the position vector is shown below:

[0214]

[0215] When the robotic arm moves at a constant speed between the two midpoints and (t) i +t acci / 2)≤t<(t i+1 -t acc(i+1) When i = 2, 3, ..., n-1, the formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0216] a real =0

[0217] The formula for the real-time value of the velocity vector is shown below:

[0218] v real =v n-1

[0219] The formula for the real-time value of the position vector is shown below:

[0220] θ real =θ n-1 +v n-1 ·(tt n-1 )

[0221] When the robotic arm is in the deceleration phase from the last intermediate path point to the end path point and (t) n -t accn )≤t≤t n The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows:

[0222] a real =a end

[0223] The formula for the real-time value of the velocity vector is shown below:

[0224] v real =v n-1 +a end ·(tt n +t accn )

[0225] The formula for the real-time value of the position vector is shown below:

[0226]

[0227] Specifically, the preset sampling time t < t acc1 The formula for the real-time value of the position vector is as follows:

[0228]

[0229] Preset sampling time t acc1 ≤t<(t2-t acc2 When / 2), the formula for the real-time value of the position vector is as follows:

[0230] θ real =θ1+v a ·t

[0231] Preset sampling time (t2-t) acc2 / 2)≤t<(t2+t acc2 When / 2), the formula for the real-time value of the position vector is as follows:

[0232]

[0233] Preset sampling time (t2+t) acc2 / 2)≤t<(t3-t acc3 When / 2), the formula for the real-time value of the position vector is as follows:

[0234] θ real =θ2+v b ·(t-t2)

[0235] Preset sampling time (t3-t) acc3 When t ≤ t3, the formula for the real-time value of the position vector is as follows:

[0236]

[0237] Based on the above formula, we can obtain Figure 4 The joint position curves shown indicate that within the joint space, each axis transitions with 20 sampling points at the path points without abrupt changes, and the slope remains constant between the path points, demonstrating excellent stability. Figure 5 For the same path points and the same arrival time, the result of fifth-order polynomial programming is used. In the figure, the rotation angle of joint 6 changes abruptly from 3.12 rad to 2.35 rad at the initial stage, and at the same time, it changes abruptly from 2.06 rad to 2.41 rad when transitioning to the second path point. The instantaneous changes are large, and there will be obvious motor vibration. The slope of the curve changes constantly between path points, and the speed of the joint motor is in dynamic change.

[0238] Specifically, the preset sampling time t < t acc1 The formula for the real-time value of the velocity vector is as follows:

[0239] v real =[v real1 v real2 v real3 v real4 v real5 v real6 ] = a start ·t

[0240] Preset sampling time (t2-t) acc2 / 2)≤t<(t2+t acc2 When / 2), the formula for the real-time value of the velocity vector is as follows:

[0241] v real =v a

[0242] Preset sampling time (t2+t) acc2 / 2)≤t<(t3-t acc3 When / 2), the formula for the real-time value of the velocity vector is as follows:

[0243] v real =v b

[0244] Preset sampling time (t3-t) acc3 When t ≤ t3, the formula for the real-time value of the velocity vector is as follows:

[0245] v real =v b +a end ·(t-t3+t acc3 )

[0246] Based on the above formula, we can obtain Figure 6 The joint velocity curve shown consists of five parts: initial acceleration segment, first uniform speed segment, intermediate variable speed segment, second uniform speed segment, and final deceleration segment. During the process, the initial and final speeds of the motor are both 0. The speed is stable during the uniform speed segment. Constrained by the given Cartesian space speed, the maximum positive speed of the six joints is 0.232 rad / s, the maximum negative speed is -0.319 rad / s, and the response between the six axes is consistent. Figure 7 The result of the fifth-order polynomial programming only guarantees the constraint that the speed at each path point is 0. The speed increases and then decreases between path points, and there is no constant speed segment. Compared with the planning in Example 1, the maximum positive speed of the joint motor is 0.421 rad / s and the maximum negative speed is -0.6 rad / s. It requires a larger output torque and the joint speed variation range is larger.

[0247] Preset sampling time t < t acc1 The formula for the real-time value of the joint acceleration vector is as follows:

[0248] a real =0

[0249] Preset sampling time (t2-t) acc2 / 2)≤t<(t2+t acc2 At / 2), the formula for the real-time value of the joint acceleration vector is as follows:

[0250] a real =a middle

[0251] Preset sampling time (t2+t) acc2 / 2)≤t<(t3-t acc3 At / 2), the formula for the real-time value of the joint acceleration vector is as follows:

[0252] a real =0

[0253] Preset sampling time (t3-t) acc3 When t ≤ t3, the formula for the real-time value of the joint acceleration vector is as follows:

[0254] a real =a end

[0255] Based on the above formula, we can obtain Figure 8 The joint acceleration curve shown has three acceleration-deceleration processes, the response time of which is determined by the given acceleration-deceleration time. Acceleration is completed at path point 1 after passing through 41 sampling points, acceleration is completed at path point 2 after passing through 79 sampling points, and deceleration is completed at path point 3 after passing through 40 sampling points. The three acceleration-deceleration processes can be completed in a short time. The maximum positive acceleration is 0.813 rad / s2, the maximum negative acceleration is -0.53 rad / s2, and the other two are uniform acceleration processes with zero acceleration. Figure 9 The result of the fifth-order polynomial programming can be found that, even when the constraint condition of zero acceleration at path points is met, there are still cases where the acceleration is zero. In the first path segment, 200 sampling points have positive acceleration and 200 sampling points have negative acceleration. In the second path segment, 639 sampling points have positive acceleration and 639 sampling points have negative acceleration. Compared with the planning result of the embodiment of this application, the acceleration of the fifth-order polynomial programming changes continuously throughout the motion process. The joint motors need to constantly change the output torque to respond, which inevitably leads to shaking problems and affects the stability of the drone.

[0256] Combination Figures 4 to 9Analysis of the curves for the position, velocity, and acceleration of each joint shows that the embodiment of this application exhibits a relatively fast speed response, smooth transition of the trajectory at intermediate points, and uniform motion across each path segment with stable changes and no significant positional jitter or shift. Therefore, it has a minimal impact on the UAV balance control system. It should be noted that this embodiment is illustrated using three path points; the number of path points is not intended to limit the scope of protection of this embodiment.

[0257] like Figure 2 As shown, this application embodiment provides a motion trajectory planning device for a robotic arm, including a distance calculation module 201, a time calculation module 202, a speed calculation module 203, and a trajectory planning module 204, wherein:

[0258] The distance calculation module 201 is configured to obtain the positions of n path points, including the initial path point, the final path point, and n-2 intermediate path points; and to calculate the absolute distance from each path point to the initial path point and the total absolute distance based on the positions of the n path points.

[0259] The time calculation module 202 is configured to set a preset spatial speed, calculate the total time spent based on the preset spatial speed and the total absolute distance; calculate the time to reach each path point based on the absolute distance from each path point to the initial path point, the total absolute distance, and the total time spent; set a preset speed change time, and calculate the speed change time required for each path point based on the preset speed change time.

[0260] The speed calculation module 203 is configured to calculate the angle vectors of the m-axis motors in the joint space corresponding to the n path points in the Cartesian space of the robotic arm, and the velocity and acceleration of the m-axis motors in the uniform segment and variable segment of each path segment.

[0261] The trajectory planning module 204 is configured to preset the sampling time, calculate the real-time values ​​of the joint acceleration vector, velocity vector and position vector of the robotic arm at each time point according to the preset sampling time, and plan the motion trajectory of the robotic arm according to the real-time values ​​of the joint acceleration vector, velocity vector and position vector of the robotic arm at each time point.

[0262] The motion trajectory planning device for the robotic arm in this application embodiment can be a computer device or a component within the computer device, such as an integrated circuit or a chip. The computer device can be a terminal or other devices besides a terminal. For example, the computer device can be a mobile phone, tablet computer, laptop computer, PDA, in-vehicle computer device, mobile internet device (MID), ultra-mobile personal computer (UMPC), netbook, or personal digital assistant (PDA), etc., and can also be a server, network attached storage (NAS), personal computer (PC), etc. This application embodiment does not specifically limit the specific implementation.

[0263] The motion trajectory planning device for the robotic arm provided in this application embodiment can achieve… Figure 1 The various processes implemented in the embodiment of the motion trajectory planning method for the robotic arm will not be described in detail here to avoid repetition.

[0264] This application also provides a computer device, such as... Figure 3 As shown, the computer device includes a processor 301 and a memory 302. The memory 302 stores programs or instructions that can run on the processor 301. When the program or instructions are executed by the processor 301, they implement the various steps of the above-mentioned robotic arm motion trajectory planning method and achieve the same technical effect. To avoid repetition, they will not be described in detail here.

[0265] It should be noted that the computer device in this application embodiment includes the mobile computer device and the non-mobile computer device described above.

[0266] The memory 302 can be used to store software programs and various data. The memory 302 may primarily include a first storage area for storing programs or instructions and a second storage area for storing data. The first storage area may store the operating system, application programs or instructions required for at least one function (such as sound playback, image playback, etc.). Furthermore, the memory 302 may include volatile memory or non-volatile memory, or both. The non-volatile memory may be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. Volatile memory can be random access memory (RAM), static random access memory (SRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link dynamic random access memory (SLDRAM), and direct memory bus RAM (DRRAM). The memory 302 in this embodiment includes, but is not limited to, these and any other suitable types of memory.

[0267] Processor 301 may include one or more processing units; optionally, processor 301 integrates an application processor and a modem processor, wherein the application processor mainly handles operations involving the operating system, user interface, and applications, and the modem processor mainly handles wireless communication signals, such as a baseband processor. It is understood that the aforementioned modem processor may also not be integrated into processor 301.

[0268] This application also provides a readable storage medium storing a program or instructions. When the program or instructions are executed by a processor, they implement the various processes of the above-described robotic arm motion trajectory planning method embodiment and achieve the same technical effect. To avoid repetition, they will not be described again here.

[0269] This application also provides a chip, which includes a processor and a communication interface. The communication interface and the processor are coupled. The processor is used to run programs or instructions to implement the various processes of the above-described robotic arm motion trajectory planning method embodiments and can achieve the same technical effect. To avoid repetition, it will not be described again here.

[0270] It should be understood that the chip mentioned in the embodiments of this application may also be referred to as a system-on-a-chip, system chip, chip system, or system-on-a-chip, etc.

[0271] This application also provides a computer program product, which is stored in a storage medium and executed by at least one processor to implement the various processes of the above-described robotic arm motion trajectory planning method embodiment, and can achieve the same technical effect. To avoid repetition, it will not be described again here.

[0272] It should be noted that, in this application, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element. Furthermore, it should be noted that the scope of the methods and apparatuses in the embodiments of this application is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in the reverse order, depending on the functions involved. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. Additionally, features described with reference to certain examples may be combined in other examples.

[0273] The embodiments of this application have been described above with reference to the accompanying drawings. However, this application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of this application.

Claims

1. A method for planning a motion trajectory of a robot arm, characterized in that, The method includes: acquiring path point positions, the path point positions including an initial path point, a terminal path point and intermediate path points; calculating absolute distances of each path point to the initial path point and a total absolute distance according to the path point positions; Set a preset spatial speed, and calculate the total time spent based on the preset spatial speed and the total absolute distance; calculate the time to reach each path point based on the absolute distance from each path point to the initial path point, the total absolute distance, and the total time spent. Set a preset speed change time, and calculate the required speed change time for each path point based on the preset speed change time, including: The preset gear shift time is set as The formula of the gear shift time required by each path point is as follows: in, Indicates the preset speed change time, when hour, Indicates the acceleration time of the initial path point; when hour, Indicates the deceleration time at the end path point; when hour, This indicates the time of acceleration / deceleration at intermediate path points; The formula for the variable-speed time vector is shown below: wherein denotes the shift time vector; The computing machine arm in the Cartesian space, with The angle vector of the joint space corresponding to the path point of the The axis motor of the joint space corresponding to the path point of the The uniform speed segment and the variable speed segment acceleration of the axis motor in each path segment A preset sampling time is used to calculate the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point. The motion trajectory of the robotic arm is planned based on the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point.

2. The motion trajectory planning method for a robotic arm as described in claim 1, characterized in that, The method according to the The method according to the The method according to the set up The coordinates of the path points in Cartesian space are: ,in The formula for the absolute distance from each path point to the initial path point is as follows: Total absolute distance The formula for the distance vector is shown below: in, Indicates the path from the initial path point to the path point. absolute distance, This represents the total absolute distance from the initial path point to the final path point. This represents the distance vector.

3. The method of Claim 2, wherein, The setting of a preset spatial speed, and the calculation of the total time spent based on the preset spatial speed and the total absolute distance, includes: Set the preset space velocity as The formula for the total time spent is as follows: in, Indicates the total time spent. Indicates the preset space velocity; The formula for the time to reach each waypoint is as follows: in, Indicates reaching the waypoint Time; The formula for the time vector is shown below: in, Represents the time vector.

4. The motion trajectory planning method for a robotic arm as described in claim 1, characterized in that, Before calculating the required shift time for each path point based on the preset shift time, a time constraint condition is determined. The time constraint condition includes that the shift time between two path points is less than the total time spent between the two path points, as shown in the following formula: After satisfying the time constraint conditions, calculate the velocity of the uniform segment and the acceleration of the variable segment for each path segment.

5. The motion trajectory planning method for a robotic arm as described in claim 1, characterized in that, The computational robotic arm is in Cartesian space, and The joint space corresponding to each path point The angle vector of the individual axis motor and The speed of the single-axis motor in the constant speed segment and the acceleration in the variable speed segment of each path segment include: The robotic arm in Cartesian space is obtained through inverse kinematics. The joint space corresponding to each path point Angle vector of the axis motor The formula is shown below: in ; The path points include Each path segment The speed formulas for the uniform speed segments of each axis motor are shown below: in, The speed change segments of each axis motor in each path segment include the acceleration segment from the initial path point to the intermediate path point, the deceleration segment from the intermediate path point to the end path point, and the speed change segment between the first intermediate path point and the last intermediate path point. The acceleration vector formula for the individual axis motor during the acceleration segment from the initial path point to the intermediate path point is shown below: ； The formula for the acceleration vector of a single-axis motor during the deceleration phase from the intermediate path point to the final path point is shown below: ； The acceleration vector formula for a single-axis motor during the speed change segment between the first intermediate path point and the last intermediate path point is shown below: ； in, .

6. The motion trajectory planning method for a robotic arm as described in claim 5, characterized in that, The preset sampling time is used to calculate the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point, including: The preset sampling time is set as follows: ; When the robotic arm is in the acceleration phase from the initial path point to the first intermediate path point and The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows: The formula for the real-time value of the velocity vector is shown below: The formula for the real-time value of the position vector is shown below: When the robotic arm moves at a constant speed from the initial path point to the first intermediate path point and The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows: The formula for the real-time value of the velocity vector is shown below: The formula for the real-time value of the position vector is shown below: When the robotic arm moves in the speed-changing segment between the two midpoints and The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows: The formula for the real-time value of the velocity vector is shown below: The formula for the real-time value of the position vector is shown below: When the robotic arm moves at a constant speed between the two midpoints and The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows: The formula for the real-time value of the velocity vector is shown below: The formula for the real-time value of the position vector is shown below: When the robotic arm is in the deceleration phase from the last intermediate path point to the end path point and The formula for the real-time value of the joint acceleration vector of the robotic arm is as follows: The formula for the real-time value of the velocity vector is shown below: The formula for the real-time value of the position vector is shown below:

7. A motion trajectory planning device for a robotic arm, characterized in that, The device includes: The distance calculation module is configured to obtain... The path point locations, the The path point locations include the initial path point, the final path point, and... There are intermediate path points; according to the above Calculate the absolute distance from each path point to the initial path point and the total absolute distance; The time calculation module is configured to set a preset spatial speed, calculate the total time spent based on the preset spatial speed and the total absolute distance; calculate the time to reach each path point based on the absolute distance from each path point to the initial path point, the total absolute distance, and the total time spent; set a preset speed change time, and calculate the required speed change time for each path point based on the preset speed change time, including: Set the preset speed change time to The formula for the required change-of-speed time at each path point is as follows: in, Indicates the preset speed change time, when hour, Indicates the acceleration time of the initial path point; when hour, Indicates the deceleration time at the end path point; when hour, This indicates the time of acceleration / deceleration at intermediate path points; The formula for the variable-speed time vector is shown below: in, Represents the variable-speed time vector; the velocity calculation module is configured to calculate the speed of the robotic arm in Cartesian space, and... The joint space corresponding to each path point The angle vector of the individual axis motor and The speed and acceleration of the motor in the constant speed section and the variable speed section of each path segment; The trajectory planning module is configured to preset a sampling time, calculate the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point based on the preset sampling time, and plan the motion trajectory of the robotic arm based on the real-time values ​​of the joint acceleration vector, velocity vector, and position vector of the robotic arm at each time point.

8. A motion trajectory planning system for a robotic arm, the system comprising a processor and a memory, wherein the memory stores a computer program, characterized in that, The computer program is loaded and executed by the processor to implement the motion trajectory planning method for the robotic arm as described in any one of claims 1 to 6.

9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it is used to implement the motion trajectory planning method for the robotic arm as described in any one of claims 1 to 6.

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