Feasible trajectory implementation system and method for aerial manipulator system dynamics

Abstract

The application discloses a system and method for realizing a dynamic feasible trajectory of an aerial manipulator system, which comprises a system state trajectory inversion module and a dynamic feasible trajectory optimization solving module, wherein the system state trajectory inversion module obtains an aerial manipulator system state trajectory through system kinematic inverse solution according to a specified actuator task trajectory; and the dynamic feasible trajectory optimization solving module obtains a dynamic feasible trajectory through numerical optimization solving according to system differential algebraic equation constraints.The application realizes the conversion of dynamic feasibility into smoothness requirements of flat outputs by determining one-to-one mapping from system flat outputs to full states of actuators, and determines the system dynamic feasible state trajectory by solving an optimization problem.

Description

Technical Field

[0001] This invention relates to a technology in the field of robot control, specifically a system and method for realizing the dynamic feasible trajectory of an aerial robotic arm system based on actuator pose inverse mapping. Background Technology

[0002] Most existing aerial robotic arms are combinations of rotary-wing drones and robotic arms. However, quadcopter drones have underactuated characteristics, and there are complex dynamic coupling effects between the robotic arm and the drone. This means that any specified actuator trajectory may not be dynamically feasible for the entire system, thus affecting the accuracy of actuator operation control. Summary of the Invention

[0003] This invention addresses the limitations of existing technologies that, when inversely solving the state trajectory of an aerial robotic arm system based on the actuator's task trajectory, can only achieve kinematic inverse solving at the position and attitude level, or achieve dynamic inverse solving at the dynamic level at the cost of sacrificing the actuator's positional degrees of freedom. It proposes a system and method for realizing the dynamically feasible trajectory of an aerial robotic arm system. By determining a one-to-one mapping from the system's flat output to the actuator's full state, the smoothness requirement of converting dynamic feasibility into a flat output is achieved. Furthermore, the dynamically feasible state trajectory of the system is determined by solving an optimization problem.

[0004] This invention is achieved through the following technical solution:

[0005] This invention relates to a system for generating a dynamically feasible trajectory for an aerial robotic arm system, comprising: a system state trajectory inversion module and a dynamically feasible trajectory optimization solution module, wherein: the system state trajectory inversion module obtains the state trajectory of the aerial robotic arm system by inverse kinematics of the system based on a specified actuator task trajectory; the dynamically feasible trajectory optimization solution module obtains the dynamically feasible trajectory by numerical optimization solution based on the system differential algebraic equation constraints.

[0006] The system state trajectory inversion module includes: a rotation matrix reparameterization unit, a robotic arm joint rotation angle inversion unit, a UAV yaw angle inversion unit, and a system center of mass position inversion unit. Specifically: the rotation matrix reparameterization unit describes the body rotation matrix using an angular axis description method, employing the lift direction vector and the rotation angle around that direction; the robotic arm joint rotation angle inversion unit and the UAV yaw angle inversion unit calculate the desired rotation angle of the robotic arm joint and the desired yaw angle of the UAV based on the rotation matrix equation formed by the actuator's desired attitude and the body's attitude, respectively, by analogy to solving Euler angles from a known rotation matrix; the system center of mass position inversion unit obtains the desired position of the aerial robotic arm's center of mass by solving the kinematic equation formed by the actuator's pose and the system center of mass based on the desired rotation angle of the robotic arm joint.

[0007] The aforementioned dynamic feasible trajectory optimization solution module includes: a system differential algebraic equation constraint unit, a lift vector polar coordinate parameterization unit, a polar coordinate describing angle parameterization unit, and a dynamic feasible trajectory solution unit based on numerical optimization. Specifically: the system differential algebraic equation constraint unit, based on the underactuated characteristics of the quadrotor UAV and the relative motion relationship between the manipulator and the system, derives the differential algebraic equations that the aerial manipulator must satisfy through dynamic modeling and analysis; the lift vector polar coordinate parameterization unit, based on the motion characteristics of the lift vector constrained on a three-dimensional unit sphere, calculates the dynamic feasible trajectory solution using the equations in three-dimensional spherical coordinates. The azimuth and polar angles are used to parameterize the trajectory of motion in the direction of lift. The polar coordinate description angle parameterization unit, based on the general representation ability of trigonometric series for general functions, parameterizes the azimuth and polar angle trajectories through the amplitude and angular frequency of finite-order trigonometric functions. The dynamic feasible trajectory solving unit based on numerical optimization, according to the dynamic feasibility constraints represented by the system's differential algebraic equations, generates a system dynamic feasible trajectory that satisfies the constraints of the differential algebraic equations by first searching in the low-dimensional parameter space formed by the trigonometric series and then fine-tuning in the high-dimensional parameter space formed by the entire lift vector trajectory.

[0008] Technical effect

[0009] This invention establishes a theoretical mapping relationship from the six-DOF pose of the actuator to the full state of the aerial robotic arm system. It proposes a method to generate the full-state trajectory of the system by parameterizing the lift direction trajectory represented in polar coordinates through a trigonometric series and then solving the differential-algebraic equations through numerical optimization. Compared with existing technologies, this invention can achieve high-precision numerical solutions for the full-state trajectory of the aerial robotic arm system given the six-DOF pose trajectory of the actuator, and ensures that the obtained trajectory is dynamically feasible for the entire aerial robotic arm system, thus providing a reliable reference trajectory for the precise control of the actuator. Attached Figure Description

[0010] Figure 1 This is a schematic diagram of the system of the present invention;

[0011] Figure 2 A coordinate system diagram describing the aerial robotic arm;

[0012] Figure 3 A schematic diagram illustrating the inversion of the full-state trajectory of an aerial robotic arm based on the actuator's task trajectory;

[0013] In the figure: the blue dashed line is the full-state trajectory of the system, including the UAV pose and the joint rotation of the robotic arm; the blue curved surface is the full-state space of the aerial robotic arm system; the orange dashed line is the task trajectory, including the actuator pose; the orange plane is the actuator pose space; and the red arrow indicates the inversion process from the actuator pose task trajectory to the full-state trajectory of the system.

[0014] Figure 4 A schematic diagram of the system state trajectory obtained under the pose trajectory of the figure-eight shaped rotary actuator;

[0015] In the figure: (a) is the trajectory of the actuator position and the system center of mass position, and (b) is the trajectory of the robot arm joint and the body yaw.

[0016] Figure 5 A schematic diagram of the trajectory of the lift direction of the aircraft under the mission trajectory of the figure-eight actuator;

[0017] Figure 6 A schematic diagram of the complete trajectory of the aerial robotic arm in three-dimensional space under the task trajectory of the figure-eight actuator;

[0018] Figure 7 A schematic diagram of the system state trajectory obtained under the pose trajectory of the circular position and attitude rotary actuator;

[0019] In the figure: (a) is the trajectory of the actuator position and the system center of mass position, and (b) is a schematic diagram of the trajectory of the robot arm joint and the body yaw.

[0020] Figure 8 A schematic diagram of the trajectory of the airframe's lift direction under the mission trajectory of the circular actuator;

[0021] Figure 9 This is a schematic diagram of the complete trajectory of the aerial robotic arm in three-dimensional space under the task trajectory of the circular actuator. Detailed Implementation

[0022] like Figure 1 As shown in this embodiment, a method for realizing the dynamically feasible trajectory of an aerial robotic arm system is proposed. By determining a one-to-one mapping from the system's flat output to the actuator's full state, the dynamic feasibility of the actuator trajectory is transformed into constraints of the smoothness of the flat output and the system's differential-algebraic equations. Then, an optimization problem is solved to determine the dynamically feasible state trajectory of the system. Specifically, this includes:

[0023] Step 1) Establish the coordinate system of the aerial robotic arm: To accurately describe the motion and state of the aerial robotic arm, such as... Figure 2 A suitable spatial coordinate system is established as shown. The motion state of the quadcopter drone, robotic arm, and aerial robotic arm system is described in the state space, and the motion state of the robotic arm actuator is described in the task space.

[0024] The spatial coordinate system mentioned includes: an inertial coordinate system. Body coordinate system And the robotic arm A linkage coordinate system All coordinate systems follow the right-hand rule.

[0025] The state space described in the text describes the motion state of the quadcopter UAV, including: the position of the UAV's center of mass. From an inertial frame of reference To machine system rotation matrix and the yaw angle of the aircraft The description of the robotic arm's motion includes: the angle of each joint. It is confined to a unit ring. The combined vector representation of all joint rotation angles during the upper motion. ,in: of The Cartesian product; describing the motion state of an aerial robotic arm system includes: the position of the system's center of mass. .

[0026] The number of robotic arm joints in this embodiment .

[0027] The motion state of the robotic arm actuator described in the task space includes: actuator position. and attitude rotation matrix .

[0028] Step 2) as Figure 3 As shown, the process of inverting the full-state trajectory of an aerial robotic arm system from the actuator's mission trajectory is illustrated. The objective is to deduce, based on the given actuator pose, the UAV pose and robotic arm joint angles that conform to the dynamics feasibility of the aerial robotic arm. Specifically, after reparameterizing the body rotation matrix using the body direction vector and the rotation angle around the axis, given the actuator pose and lift direction vector, inverse mapping expressions for the robotic arm joint angles, body yaw angles, and the system's center of mass position are constructed sequentially, including:

[0029] 2.1 Reparameterization of the body rotation matrix: using the body lift direction vector and the angle of rotation about this axis The rotation matrix of the machine body is described by the angular axis as the independent variable. The rotation process from the slave system to the inertial frame can be decomposed into two steps, specifically: ,in: Indicates the first step of the winding system Axis rotation Angle rotation matrix; This represents the rotation matrix that, based on the first rotation, rotates to the inertial frame.

[0030] 2.2 Given the actuator pose and lift direction vector, reconstruct the joint rotation angle of the robotic arm: The product of the two pose matrices is simply (The rest of the text is missing). The direction of lift can be determined by following the steps of solving for Euler angles using a known rotation matrix. Rotation angle The joints of the two robotic arms rotate at angles. and ,in: for Transpose of a matrix; The actuator attitude rotation matrix; For matrix The corresponding number in the middle OK The elements of the column.

[0031] 2.3 Given the actuator pose and lift direction vector, solve for the airframe yaw angle by inverse equation: The product of the two rotation matrices is simply denoted as... Then, by analogy with the process of solving for Euler angles from a known rotation matrix, the yaw angle of the aircraft can be determined. ,in: For matrix The corresponding number in the middle OK The elements of the column.

[0032] 2.4 Given the actuator pose and lift direction vector, solve for the system's center of mass position: Since the joint angles of the robotic arm have already been determined, from... arrive rotation matrix And from arrive rotation matrix All are known. Simultaneously, from the center of mass of the first link of the robotic arm to... The offset vector in Expression in the system ,from arrive The offset vector in Expression in the system From the center of mass of the second connecting rod to The offset vector in Expression in the system ,from arrive The offset vector in Expression in the system These quantities are only related to the length of the robotic arm links and are all known quantities. Therefore, the position of the system's center of mass can be solved inversely based on the kinematic relationship of the robotic arm: ,in: The location of the system's centroid; For actuator position; The total mass of the system; For drone quality; Let be the mass of the two connecting rods.

[0033] In this embodiment, the mass of the quadcopter drone is: kg, the mass of the connecting links of the two robotic arms is kg kg, the length of the two connecting rods is m.

[0034] Step 3) The pose trajectory of the actuator with all six degrees of freedom, including: actuator position trajectory parameterization and actuator attitude Euler angle trajectory parameterization, specifically including:

[0035] 3.1 According to Euler angle sequence construction of actuator attitude rotation matrix ,in: Indicates circling Direction rotation Angle rotation matrix; Indicates circling Direction rotation Angle rotation matrix; Indicates circling Direction rotation Rotation matrix of angle.

[0036] 3.2 Actuator position trajectory parameterization, including: figure-eight trajectory and circular trajectory .

[0037] In this embodiment, the trajectory radius is m, angular velocity is rad / s.

[0038] 3.3 Actuator attitude trajectory parameterization, that is, specifying the three Euler angle trajectories as uniform rotation: taking the yaw direction as an example, specifying... ,in: Angular velocity of rotation The initial angular position is used as the parameterization point, and the Euler angles in the other two directions are parameterized in a similar manner. , The corresponding rotational angular velocity is The initial angle position is .

[0039] In this embodiment, the angular velocities corresponding to the Euler angles in the three directions of the actuator are: rad / s, initial angle position is rad.

[0040] Step 4) Add system differential-algebraic equation constraints, specifically including:

[0041] 4.1 The direction of lift should always be perpendicular to the plane of the aircraft and propeller, therefore it must satisfy the following relationship with the acceleration of the system's center of mass: ,in: It refers to a unit sphere in three-dimensional space. It is the acceleration due to gravity. Inertial frame Orientation coordinate basis, The acceleration of the system's center of mass.

[0042] Gravitational acceleration in this embodiment m / s 2 .

[0043] 4.2 The influence of the robotic arm's motion on the quadcopter drone's motion, i.e., the position of the system's center of mass. With actuator position The following conditions must be met: This is consistent with the inverse solution process of the system's centroid location in step 2).

[0044] Step 5) Parameterization of the lift direction trajectory, specifically: using the azimuth angle and polar angle Parameterized lift direction vector Using trigonometric series to parameterize the azimuth and polar angles, and using sine and cosine basis parameters to parameterize the trajectory of the angles: , ,in: The fundamental frequency of the trigonometric series corresponding to the polar angle and azimuth angle; and These are the cosine and sine amplitudes of the trigonometric series corresponding to the polar angle and azimuth angle, respectively; parameter set. These are the parameters to be optimized in the angle trajectory.

[0045] In this embodiment, the order of the trigonometric series is determined as follows: .

[0046] Step 6) Construct a numerical optimization problem and solve for the feasible trajectory of the system dynamics, specifically including:

[0047] The numerical optimization problem constructed in 6.1 is as follows: ,in: For the system trajectory throughout the entire time period The number of sampling points within; Sampling time; The acceleration of the system's center of mass is determined by the position of the system's center of mass. The system's centroid position is approximated by using a second-order numerical difference method. It is obtained from the inverse solution in step 2).

[0048] 6.2 After solving the lift direction trajectory that meets the system differential algebra constraints in step 4) through numerical optimization, the complete state trajectory of the entire aerial robotic arm system is determined according to the inverse solution method from actuator pose to system state in step 2).

[0049] In this embodiment, the total duration is A total of samples were taken on the trajectory of s. A discrete point.

[0050] To evaluate the dynamic feasibility of obtaining the complete system trajectory, the cost of the objective function of the optimization problem established in step 6) is evaluated. At the same time, the error between the actual actuator pose trajectory obtained by forward calculation of the complete system trajectory obtained by optimization and the specified actuator pose trajectory is compared.

[0051] The evaluation of the cost value of the objective function of the optimization problem is based on the following formula, which calculates the dynamic error between the actual lift direction and the theoretically feasible lift direction. ,in: The trajectory of the lift direction obtained in step 6); To obtain the acceleration of the system's center of mass.

[0052] The actual actuator pose trajectory obtained by forward calculation based on the optimized system complete trajectory, and the error between this trajectory and the pre-specified actuator pose trajectory, can be used to directly calculate the actuator position error. The error of the actuator attitude rotation matrix is ,in: To specify the actuator position and attitude trajectory; The actual actuator position and attitude trajectory are calculated based on the lift direction obtained from the solution; Let be the trace of the matrix.

[0053] Through specific simulation experiments, in the MATLAB numerical simulation environment, for the two scenarios specified in step 3), the numerical optimization problem constructed in step 6) was established. Using the parameterization technique of trigonometric series for the lift direction vector in step 5), the lift direction trajectory and the system's center of mass motion trajectory, satisfying the system's differential algebraic equation constraints in step 4), were solved. Finally, based on step 2), given the lift direction vector and the actuator's full pose trajectory, the motion trajectory of the entire aerial robotic arm system in the state space was obtained. Scenario 1: The actuator position trajectory is a figure-eight shaped trajectory, and the actuator attitude trajectory is a rotation matrix generated by three uniformly rotating Euler angles. Scenario 2: The actuator position trajectory is a circular "V" shaped trajectory, and the actuator attitude trajectory is a rotation matrix generated by three uniformly rotating Euler angles.

[0054] In the embodiment of scenario 1: Based on the present invention, the full-state motion trajectory of the aerial robotic arm under the figure-eight task trajectory is obtained, wherein: as follows Figure 4 As shown in (a), the blue solid line represents the trajectory of the quadcopter UAV's center of mass, and the black dashed line represents the trajectory of the aerial robotic arm system's center of mass. The three figures from left to right correspond to... The location and trajectory of the channel. For example... Figure 4 As shown in (b), from left to right, these are the yaw angles of the quadcopter drone and the angular trajectories corresponding to the joint rotation angles of the two robotic arms. Figure 5 As shown, the green dot represents the initial direction of lift for the quadcopter drone, the red dot represents the final direction of the trajectory, and the blue solid line on the unit sphere represents the evolution of the lift direction during the entire aerial robotic arm movement. Its movement is consistent with the figure-eight trajectory. Figure 6 As shown, the blue curve represents the trajectory of the quadcopter drone's center of mass, the black dashed line represents the trajectory of the aerial robotic arm system's center of mass, and the red solid line represents the trajectory of the actuators.

[0055] In the embodiment of Scenario 2: Based on the present invention, the full-state motion trajectory of the aerial robotic arm under a circular task trajectory is obtained. The trajectory has the same meaning as in Scenario 1, such as... Figure 7 As shown in (a), the blue solid line and the black dashed line represent the motion trajectories of the center of mass positions of the quadcopter drone and the aerial robotic arm system, respectively. The three figures from left to right are respectively the corresponding... The location and trajectory of the channel. For example... Figure 7 As shown in (b), from left to right, these are the angular trajectories corresponding to the yaw angle of the UAV and the joint rotation angle of the robotic arm. Figure 8 As shown, the green and red dots represent the starting and ending directions of the trajectory, and the solid blue line on the sphere represents the evolution of the lift direction. Its motion is consistent with the motion pattern of a circular trajectory. Figure 9 As shown in the figure, this diagram illustrates the complete motion state of the aerial robotic arm in three-dimensional space along a circular mission trajectory.

[0056] To further evaluate the dynamic feasibility of the method of the present invention in obtaining the system state trajectory based on the actuator task trajectory, the position and attitude errors between the specified actuator pose trajectory and the actual actuator pose trajectory calculated based on the solved lift direction vector were examined. The experimental results are shown in Tables 1 and 2.

[0057] Table 1: Numerical solution accuracy of the differential-algebraic equations for the lift direction

[0058] Table 2: Pose error between actual actuator trajectory and specified trajectory

[0059] In summary, this invention can obtain a system state trajectory that conforms to the dynamic feasibility of an underactuated aerial manipulator composed of a quadcopter UAV and a multi-degree-of-freedom manipulator, given all six degrees of freedom positions and attitude trajectories of the actuator, wherein the mean square error of the lift direction vector trajectory is within 10. -15 The order of magnitude, and the mean square error between the actual actuator pose trajectory calculated from the inverted state and the specified trajectory is within 10. -6 ~10 -4 The error in the actuator attitude rotation matrix is ​​on the order of m, and is within 10. -9 ~10 -6 The magnitude indicates that the high-precision inversion of the state trajectory has good dynamic feasibility.

[0060] Specific implementations may be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of this invention. The scope of protection of this invention is defined by the claims and is not limited by specific implementations. All implementation schemes within its scope are bound by this invention.

Claims

1. A dynamically feasible trajectory generation system for an aerial robotic arm system, characterized in that, include: The system consists of a system state trajectory inversion module and a dynamic feasible trajectory optimization solution module. Specifically, the system state trajectory inversion module obtains the aerial robotic arm system state trajectory by inverse kinematics of the system based on the specified actuator task trajectory; the dynamic feasible trajectory optimization solution module obtains the dynamic feasible trajectory by numerical optimization solution based on the system differential algebraic equation constraints. The system state trajectory inversion module includes: a rotation matrix reparameterization unit, a robotic arm joint rotation angle inversion unit, a UAV yaw angle inversion unit, and a system center of mass position inversion unit. Specifically: the rotation matrix reparameterization unit describes the body rotation matrix using an angular axis description method, employing the lift direction vector and the rotation angle around that direction; the robotic arm joint rotation angle inversion unit and the UAV yaw angle inversion unit calculate the desired rotation angle of the robotic arm joint and the desired yaw angle of the UAV based on the rotation matrix equation formed by the actuator's desired attitude and the body's attitude, respectively, by analogy to solving Euler angles from a known rotation matrix; the system center of mass position inversion unit obtains the desired position of the aerial robotic arm's center of mass by solving the kinematic equation formed by the actuator's pose and the system center of mass based on the desired rotation angle of the robotic arm joint. The aforementioned dynamic feasible trajectory optimization solution module includes: a system differential algebraic equation constraint unit, a lift vector polar coordinate parameterization unit, a polar coordinate describing angle parameterization unit, and a dynamic feasible trajectory solution unit based on numerical optimization. Specifically: the system differential algebraic equation constraint unit, based on the underactuated characteristics of the quadrotor UAV and the relative motion relationship between the manipulator and the system, derives the differential algebraic equations that the aerial manipulator must satisfy through dynamic modeling and analysis; the lift vector polar coordinate parameterization unit, based on the motion characteristics of the lift vector constrained on a three-dimensional unit sphere, calculates the dynamic feasible trajectory solution using the equations in three-dimensional spherical coordinates. The azimuth and polar angles are used to parameterize the trajectory of motion in the direction of lift. The polar coordinate description angle parameterization unit, based on the general representation ability of trigonometric series for general functions, parameterizes the azimuth and polar angle trajectories through the amplitude and angular frequency of finite-order trigonometric functions. The dynamic feasible trajectory solving unit based on numerical optimization, according to the dynamic feasibility constraints represented by the system's differential algebraic equations, generates a system dynamic feasible trajectory that satisfies the constraints of the differential algebraic equations by first searching in the low-dimensional parameter space formed by the trigonometric series and then fine-tuning in the high-dimensional parameter space formed by the entire lift vector trajectory.

2. A method for generating a dynamically feasible trajectory of an aerial robotic arm system according to claim 1, characterized in that, By establishing a one-to-one mapping from the system's flat output to the actuator's full state, the dynamic feasibility of the actuator trajectory is transformed into constraints of the smoothness of the flat output and the system's differential algebraic equations. Then, the optimization problem is solved to determine the dynamically feasible state trajectory of the system.

3. The method for generating a dynamically feasible trajectory for an aerial robotic arm system according to claim 2, characterized in that, specifically... include: Step 1) Establish the coordinate system of the aerial robotic arm: In order to accurately describe the motion and state of the aerial robotic arm, a suitable spatial coordinate system is established. The motion state of the quadcopter drone, the robotic arm and the aerial robotic arm system is described in the state space, and the motion state of the robotic arm actuator is described in the task space. Step 2) The process of inverting the full-state trajectory of the aerial robotic arm system from the actuator mission trajectory is illustrated. The goal is to invert the UAV pose and robotic arm joint rotation angle that are feasible for aerial robotic arm dynamics based on the given actuator pose. Specifically, after reparameterizing the body rotation matrix using the body direction vector and the rotation angle around the axis, given the actuator pose and lift direction vector, inverse mapping expressions for the robotic arm joint rotation angle, body yaw angle and system center of mass position are constructed in sequence. Step 3) The pose trajectory of the actuator with all six degrees of freedom, including: actuator position trajectory parameterization and actuator attitude Euler angle trajectory parameterization; Step 4) Add system differential-algebraic equation constraints; Step 5) Parameterization of the lift direction trajectory; Step 6) Construct a numerical optimization problem and solve for the feasible trajectory of the system dynamics; The spatial coordinate system mentioned includes: an inertial coordinate system. Body coordinate system And the robotic arm A linkage coordinate system All coordinate systems follow the right-hand rule; The state space described in the text describes the motion state of the quadcopter UAV, including: the position of the UAV's center of mass. From an inertial frame of reference To machine system rotation matrix and the yaw angle of the aircraft The description of the robotic arm's motion includes: the angle of each joint. It is confined to a unit ring. The combined vector representation of all joint rotation angles during the upper motion. ,in: for of The des Cartesian product; describing the motion state of an aerial robotic arm system includes: the position of the system's center of mass. ; The motion state of the robotic arm actuator described in the task space includes: actuator position. and attitude rotation matrix .

4. The method for generating a dynamically feasible trajectory for an aerial robotic arm system according to claim 3, characterized in that, Step 2) specifically includes: 2.1 Reparameterization of the body rotation matrix: using the body lift direction vector and the angle of rotation about this axis The rotation matrix of the machine body is described by the angular axis as the independent variable. The rotation process from the slave system to the inertial frame can be decomposed into two steps, specifically: ,in: Indicates the first step of the winding system Axis rotation Angle rotation matrix; This represents the rotation matrix that, based on the first rotation, rotates to the inertial frame. 2.2 Given the actuator pose and lift direction vector, reconstruct the joint rotation angle of the robotic arm: The product of the two pose matrices is simply (The rest of the text is missing). The direction of lift can be determined by following the steps of solving for Euler angles using a known rotation matrix. Rotation angle The joints of the two robotic arms rotate at angles. and ,in: for Transpose of a matrix; The actuator attitude rotation matrix; For matrix The corresponding number in the middle OK Column elements; 2.3 Given the actuator pose and lift direction vector, solve for the airframe yaw angle by inverse equation: The product of the two rotation matrices is simply denoted as... Then, by analogy with the process of solving for Euler angles from a known rotation matrix, the yaw angle of the aircraft can be determined. ,in: For matrix The corresponding number in the middle OK Column elements; 2.4 Given the actuator pose and lift direction vector, solve for the system's center of mass position: Since the joint angles of the robotic arm have already been determined, from... arrive rotation matrix And from arrive rotation matrix It is known that, simultaneously from the center of mass of the first link of the robotic arm to... The offset vector in Expression in the system ,from arrive The offset vector in Expression in the system From the center of mass of the second connecting rod to The offset vector in Expression in the system ,from arrive The offset vector in Expression in the system These quantities are only related to the length of the robotic arm links and are all known quantities. Therefore, the position of the system's center of mass can be solved by inverse kinematics of the robotic arm: ,in: The location of the system's centroid; For actuator position; The total mass of the system; For drone quality; Let be the mass of the two connecting rods.

5. The method for generating a dynamically feasible trajectory for an aerial robotic arm system according to claim 3, characterized in that, Step 3) specifically includes: 3.1 According to Euler angle sequence construction of actuator attitude rotation matrix ,in: Indicates circling Direction rotation Angle rotation matrix; Indicates circling Direction rotation Angle rotation matrix; Indicates circling Direction rotation Angle rotation matrix; 3.2 Actuator position trajectory parameterization, including: figure-eight trajectory and circular trajectory ; 3.3 Actuator attitude trajectory parameterization, that is, specifying the three Euler angle trajectories as uniform rotation: taking the yaw direction as an example, specifying... ,in: Angular velocity of rotation The initial angular position is used as the parameterization point, and the Euler angles in the other two directions are parameterized in a similar manner. , The corresponding rotational angular velocity is The initial angle position is .

6. The method for generating a dynamically feasible trajectory for an aerial robotic arm system according to claim 3, characterized in that, Step 4) specifically includes: 4.1 The direction of lift should always be perpendicular to the plane of the aircraft and propeller, therefore it must satisfy the following relationship with the acceleration of the system's center of mass: ,in: It refers to a unit sphere in three-dimensional space. It is the acceleration due to gravity. Inertial frame Orientation coordinate basis, For the acceleration of the system's center of mass; 4.2 The influence of the robotic arm's motion on the quadcopter drone's motion, i.e., the position of the system's center of mass. With actuator position The following conditions must be met: This is consistent with the inverse solution process of the system's centroid location in step 2).

7. The method for generating a dynamically feasible trajectory for an aerial robotic arm system according to claim 3, characterized in that, Step 5) specifically involves: using the azimuth angle and polar angle Parameterized lift direction vector Using trigonometric series to parameterize the azimuth and polar angles, and using sine and cosine basis parameters to parameterize the trajectory of the angles: , ,in: The fundamental frequency of the trigonometric series corresponding to the polar angle and azimuth angle; and These are the cosine and sine amplitudes of the trigonometric series corresponding to the polar angle and azimuth angle, respectively; parameter set. These are the parameters to be optimized in the angle trajectory.

8. The method for generating a dynamically feasible trajectory for an aerial robotic arm system according to claim 3, characterized in that, Step 6) specifically includes: The numerical optimization problem constructed in 6.1 is as follows: ,in: For the system trajectory throughout the entire time period The number of sampling points within; Sampling time; The acceleration of the system's center of mass is determined by the position of the system's center of mass. The system's centroid position is approximated by using a second-order numerical difference method. Obtained by inverse solution in step 2); 6.2 After solving the lift direction trajectory that meets the system differential algebra constraints in step 4) through numerical optimization, the complete state trajectory of the entire aerial robotic arm system is determined according to the inverse solution method from actuator pose to system state in step 2).

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