Analysis method for operating force of tail end of under-actuated multi-degree-of-freedom mechanical arm

Abstract

The invention relates to the technical field of underactuated mechanical arm motion control, and discloses an underactuated multi-degree-of-freedom mechanical arm tail end operating force analysis method which comprises the steps that according to an underactuated multi-degree-of-freedom mechanical arm mathematical model and a two-force balance condition, constraint conditions of a mechanical arm tail end operating force component and a torque component are established; according to the constraint conditions of the tail end operating force component and the torque component of the mechanical arm, all components of the tail end operating force of the mechanical arm are solved by combining the spatial position relation between a mechanical arm mechanism and a tail end operating force vector; and calculating the tail end operating force of the mechanical arm according to each component of the tail end operating force of the mechanical arm. According to the method, the motion characteristics of the under-actuated mechanical arm can be comprehensively considered, and accurate analysis of the tail end operating force of the under-actuated multi-degree-of-freedom mechanical arm is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of motion control of an underactuated manipulator, and in particular to a method for analyzing operating force at the end of an underactuated multi-degree-of-freedom manipulator. Background Art

[0002] With its small number of actuators, underactuated multi-degree-of-freedom manipulators achieve precise control of the end-effector operating force. This design shows great potential in applications that require delicate and gentle interactions. They use coupled dynamic characteristics to maintain dynamic balance and adaptive operation when performing precision tasks such as assembly, grinding or surgery. They monitor and adjust the operating force in real time through intelligent algorithms to adapt to different environments and avoid damage. In human-machine collaboration scenarios, the flexibility and force control capabilities of underactuated manipulators enable them to naturally collaborate with human workers and improve the harmony of the workflow. In addition, they serve as important tools for exploring robot dynamics and control theory in education and research, providing innovative impetus for the development of robotics. With the advancement of technology, the comprehensive advantages of underactuated manipulators in cost-effectiveness, force control and intelligent feedback indicate that they will play a more critical role in the future of automation and intelligence.

[0003] Existing research on underactuated multi-degree-of-freedom manipulators mainly focuses on their motion control methods. However, the end-operation force of the underactuated manipulator greatly affects its operation effect on tasks and targets. Therefore, combining the motion characteristics of the manipulator and accurately evaluating the end-operation force of the underactuated multi-degree-of-freedom manipulator is a prerequisite for carrying out subsequent task motion control. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a method for analyzing the operating force of an under-actuated multi-degree-of-freedom robot end.

[0005] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0006] A method for analyzing the operating force of an underactuated multi-degree-of-freedom robot end, comprising the following steps:

[0007] According to the mathematical model of the underactuated multi-degree-of-freedom manipulator and the two-force balance condition, the constraint conditions of the manipulator end operating force component and torque component are established;

[0008] According to the constraints of the end-operating force component and torque component of the manipulator, combined with the spatial position relationship between the manipulator mechanism and the end-operating force vector, the components of the end-operating force of the manipulator are solved;

[0009] According to the components of the operating force at the end of the robot arm, the operating force at the end of the robot arm is calculated.

[0010] Preferably, the constraint condition of the operating force component of the end of the robot arm is:

[0011] The force component of the end-operation force lies in the plane formed by the underactuated joint axis and the end point.

[0012] Preferably, the constraint condition of the torque component at the end of the robot arm is:

[0013] The torque components lie in the plane that passes through the end points and is perpendicular to the joint axis.

[0014] Preferably, according to the constraints of the end-operating force component and the torque component of the manipulator, combined with the spatial position relationship between the manipulator mechanism and the end-operating force vector, the components of the end-operating force of the manipulator are solved, including:

[0015] According to the spatial position of the end of the robot and the center of mass position of the underactuated joint, the constraint relationship of each element in the end-operation force component of the robot is established based on the constraint conditions of the end-operation force component of the robot;

[0016] According to the direction of the underactuated joint axis, the constraint relationship of each element in the torque component of the end of the manipulator is established based on the constraint condition of the torque component of the end of the manipulator;

[0017] According to the constraint relationship of each element in the end-operating force component of the robot arm and the constraint relationship of each element in the end-torque component of the robot arm, the components of the end-operating force of the robot arm are solved.

[0018] Preferably, the constraint relationship between the elements in the end-operation force component of the robot arm is:

[0019] ( e p ky e z kz - e p kz e z ky ) e f ex +( e p kz e z kx - e p kx e z kz ) e f ey +( e p kx e z ky - e p ky e z kx ) e fez =0

[0020] in, e p kx , e p ky , e p kz is the direction vector of the line formed by the end effector and the center of mass of the underactuated joint, e z kx , e z ky , e z kz is the direction vector of the line formed by the joint axis and the mass center of the underactuated joint, e f ex , e f ey , e f ez is the operating force component at the end of the robot arm.

[0021] Preferably, the constraint relationship of each element in the torque component of the end of the robot arm is:

[0022] e z kx e n ex + e z ky e n ey + e z kz e n ez =0

[0023] in, e z kx , e z ky , e z kz is the direction vector of the line formed by the joint axis and the mass center of the underactuated joint, e n ex , e n ey , e n ez is the torque component at the end of the robot arm.

[0024] Preferably, calculating the end operating force of the robot arm according to each component of the end operating force of the robot arm includes:

[0025] According to the force balance relationship of the underactuated multi-DOF manipulator, the relationship between the joint force and the operating force on the end effector is established;

[0026] Calculate the torque of each joint based on the relationship between the force applied to the joint and the operating force on the end effector;

[0027] The end-of-arm operating force is calculated based on the torque of each joint.

[0028] Preferably, the relationship between the force acting on the joint and the operating force on the end effector is:

[0029]

[0030]

[0031] in, i f i and i f i+1 They act on joint J i and joints J i+1 The force in the coordinate system Σ i In the representation; i n i and i n i+1 They act on joint J i and joints J i+1 The moment of force in the coordinate system Σ i In the representation; i P i+1 is the coordinate system Σ i+1 The origin of the coordinate system Σ i In the representation; The joint is defined by the coordinate system ∑ i+1 To i The posture transformation matrix.

[0032] Preferably, the calculation method of each joint torque is:

[0033]

[0034] Among them, τ i For joint J i The torque, i n i To act on joint J i The moment of force in the coordinate system Σ i In the expression, i z i For joint J i The joint axis is in the coordinate system Σ i In the representation, T is the matrix transpose.

[0035] Preferably, the calculation method of the operating force at the end of the robot arm is:

[0036]

[0037] Among them, F e is the end-operating force of the robot arm, j τi is the i-th column element of the robot force transfer matrix, τ i For joint J i is the moment of force, and n is the degree of freedom.

[0038] The present invention has the following beneficial effects:

[0039] The present invention uses the two-force balance theorem to obtain the constraints of the end operating force component and the torque component, and then solves the spatial position relationship between the manipulator mechanism and the end operating force vector to obtain the specific expressions of the end operating force components of the underactuated manipulator, and finally obtains the output torque of each joint, thereby realizing the accurate analysis of the end operating force of the underactuated multi-degree-of-freedom manipulator. The present invention can comprehensively consider the motion characteristics of the underactuated manipulator, and can provide a basis for the design of the manipulator structure and the motion control method. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 It is a flow chart of an analysis method of the end-operation force of an under-actuated multi-degree-of-freedom manipulator;

[0041] Figure 2 This is a schematic diagram of the force balance of the end operation of the underactuated manipulator;

[0042] Figure 3 It is a schematic diagram of the possible set of end operation force components in three-dimensional space;

[0043] Figure 4 It is a schematic diagram of the possible set of end operating torque components in three-dimensional space;

[0044] Figure 5 Schematic diagram of the structure of a three-degree-of-freedom planar robotic arm with the third joint under-actuated. DETAILED DESCRIPTION

[0045] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0046] like Figure 1 As shown, an embodiment of the present invention provides a method for analyzing the operating force of an under-actuated multi-degree-of-freedom robot arm end, comprising the following steps S1 to S3:

[0047] S1. According to the mathematical model of the underactuated multi-degree-of-freedom manipulator and the two-force balance condition, the constraint conditions of the operating force component and the torque component of the end of the manipulator are established;

[0048] In an optional embodiment of the present invention, this embodiment establishes the constraint conditions of the end operating force component and the torque component of the robot arm based on the mathematical model of the under-actuated multi-degree-of-freedom robot arm and the two-force balance condition, that is, the force component of the end operating force is located in the plane formed by the under-actuated joint axis and the end point, and the torque component is located in the plane passing through the end point and perpendicular to the joint axis.

[0049] First, let’s analyze the end operating force components, such as Figure 2 shown. Figure 2 In this paper, the underactuated joint and its connected arm to the end effector are considered as a whole mechanism. Analyze its force conditions. The reaction force f′ of the external target on the end e (interacting with the end operating force) and the reaction force f′ of the previous link at the underactuated joint f They are both external forces and have an impact on the overall structure. Only these two places are acted upon by external forces. An underactuated joint cannot withstand torque, i.e., τ f = 0. To maintain the overall structure Balance, according to the two-force balance condition, we have:

[0050] f′ f -f′ e =0

[0051] Among them, f′ e is the reaction force of the external target on the end of the robot arm, which is the interaction force with the end operation force output by the robot arm; f′ f is the reaction force of the previous link on the underactuated joint.

[0052] This means that the force components will pass through the underactuated joint axis. When the robot moves in three-dimensional space, the force components can only pass through the underactuated joint axis Z. f If the force components are outside this plane, the system will be unbalanced. In three-dimensional space, the possible set of end-operation force components is as follows: Figure 3 shown.

[0053] Next, we analyze the moment component. Similar to the force component analysis, the reaction moment n′ of the external target on the end e The reaction torque n′ of the underactuated joint is f To maintain balance, according to the two-force balance condition, we have:

[0054] n′ f -n′ e =0

[0055] like Figure 4 As shown, due to the underactuated joint output torque τ f = 0, so there can be no torque component at the end point parallel to the underactuated joint axis (otherwise the mechanism is unbalanced). Therefore, the direction of the torque vector can only be in the plane passing through the end point of the manipulator and perpendicular to the joint axis.

[0056] S2. According to the constraints of the end-operating force component and the torque component of the manipulator, combined with the spatial position relationship between the manipulator mechanism and the end-operating force vector, solve the components of the end-operating force of the manipulator;

[0057] In an optional embodiment of the present invention, this embodiment solves the components of the end operating force of the manipulator according to the constraints of the end operating force component and the torque component of the manipulator, combined with the spatial position relationship between the manipulator mechanism and the end operating force vector, including:

[0058] According to the spatial position of the end of the robot and the center of mass position of the underactuated joint, the constraint relationship of each element in the end-operation force component of the robot is established based on the constraint conditions of the end-operation force component of the robot;

[0059] According to the direction of the underactuated joint axis, the constraint relationship of each element in the torque component of the end of the manipulator is established based on the constraint condition of the torque component of the end of the manipulator;

[0060] According to the constraint relationship of each element in the end-operating force component of the robot arm and the constraint relationship of each element in the end-torque component of the robot arm, the components of the end-operating force of the robot arm are solved.

[0061] In this embodiment, the end operating force of the robot arm is:

[0062]

[0063] According to the theorem, the operating force needs to meet two conditions: 1) the force component can only be located in the plane formed by the underactuated joint axis and the end point; 2) the torque component can only be located in the plane passing through the end point and perpendicular to the joint axis. Next, the constraint relationship between the components of the operating force in the above formula is established by these two constraints.

[0064] For an n-DOF underactuated multi-DOF manipulator, assume that the kth joint is underactuated and its axis is in its own coordinate system Σ k Zhongwei k z k =[0 0 1] T , then its coordinate system Σ e In:

[0065]

[0066] in, Represented by the coordinate system ∑ i To i+1 The posture transformation matrix; Represented by the coordinate system ∑ n To the end effector coordinate system∑ e The posture transformation matrix.

[0067] The center of mass of the underactuated joint is Σ k Zhongwei k p k , then it is in Σ e In:

[0068]

[0069] in, Represented by the coordinate system ∑ k To k+1 The pose transformation matrix; Represented by the coordinate system ∑ n To the end effector coordinate system∑ e The pose transformation matrix.

[0070] Assume that the center of mass of the end effector is located in its own coordinate system Σ e Therefore, two non-parallel straight lines can be found on the plane formed by the end effector and the underactuated joint axis, namely the straight line formed by the joint axis and the center of mass of the end effector and the underactuated joint, and their direction vectors are ( e z kx , e z ky , e z kz )and( e p kx , e p ky , e p kz ). Because the force vector e f e =[ e f ex , e f ey , e f ez ] T Also on this plane, so the three vectors are coplanar, that is, the mixed product is zero:

[0071]

[0072] Simplifying the above formula, the constraint relationship of each element in the force component can be obtained as follows:

[0073] (e p ky e z kz - e p kz e z ky ) e f ex +( e p kz e z kx - e p kx e z kz ) e f ey +( e p kx e z ky - e p ky e z kx ) e f ez =0

[0074] in, e p kx , e p ky , e p kz is the direction vector of the line formed by the end effector and the center of mass of the underactuated joint, e z kx , e z ky , e z kz is the direction vector of the line formed by the joint axis and the mass center of the underactuated joint, e f ex , e f ey , e f ez is the operating force component at the end of the robot arm.

[0075] Due to the moment component e n e =[ e n ex , e n ey , e n ez ]′ is on the plane passing through the end effector and perpendicular to the underactuated joint axis, so the torque component is perpendicular to the underactuated joint axis. Therefore, the dot product of the torque component and the direction vector of the underactuated joint axis is zero, that is:

[0076] [e z kx , e z ky , e z kz ][ e n ex , e n ey , e n ez ] T =0

[0077] Simplifying the above formula, the constraint relationship of each element in the moment component is obtained as follows:

[0078] e z kx e n ex + e z ky e n ey + e z kz e n ez =0

[0079] The above obtains the end operating force There are two constraint relationships between the components, so there are four independent components (two force components and two moment components). By arbitrarily setting four variables, the expressions of all components of the operating force can be obtained.

[0080] S3. Calculate the end operating force of the robot arm according to the components of the end operating force of the robot arm.

[0081] In an optional embodiment of the present invention, the present embodiment calculates the end-of-arm operating force according to each component of the end-of-arm operating force, including:

[0082] According to the force balance relationship of the underactuated multi-DOF manipulator, the relationship between the joint force and the operating force on the end effector is established;

[0083] Calculate the torque of each joint based on the relationship between the force applied to the joint and the operating force on the end effector;

[0084] The end-of-arm operating force is calculated based on the torque of each joint.

[0085] For an n-DOF manipulator, based on the force balance theorem, we have:

[0086] i f i = i f i+1

[0087] in i = i n i+1 + i P i+1 × i f i+1

[0088] in, i f i and i f i+1 They act on J i and J i+1 The force in the coordinate system Σ i In the representation; i n i and i n i+1 They act on J i and J i+1 The moment of force in the coordinate system Σ i In the representation; i P i+1 is the coordinate system Σ i+1 The origin of the coordinate system Σ i Representation in; i=1,2,...,n.

[0089] for i f i+1 and i n i+1 Convert them into their own coordinate system and rewrite the above formula as follows:

[0090]

[0091]

[0092] in, i f i and i f i+1 They act on joint J i and joints J i+1 The force in the coordinate system Σ i In the representation; i n i and i n i+1 They act on joint J i and joints J i+1 The moment of force in the coordinate system Σ i In the representation; i P i+1 is the coordinate system Σ i+1 The origin of the coordinate system Σ i In the representation; The joint is defined by the coordinate system ∑ i+1 Toi The posture transformation matrix.

[0093] For joints J i , except for the torque around the joint axis, the force and torque components in other directions are borne by the robot mechanism. Therefore, the joint torque is:

[0094]

[0095] Among them, τ i For joint J i The torque, i n i To act on joint J i The moment of force in the coordinate system Σ i In the expression, i z i For joint J i The joint axis is in the coordinate system Σ i In the representation, T is the matrix transpose.

[0096] Considering the robot arm as a whole, the relationship between the force acting on J1 and the operating force on the end effector is:

[0097]

[0098]

[0099] in, e f e =[ e f ex , e f ey , e f ez ]and e n e =[ e n ex , e n ey , e n ez ] are the end operating force component and torque component respectively;

[0100] The torque of J1 is:

[0101]

[0102] in, 1 z1=[0 0 1] T .

[0103] The force analysis of other joints is similar to the above steps, and the active joint torques τ1, τ2, ..., τ are obtained in turn. n, can be substituted into the following formula to obtain the end operating force:

[0104]

[0105] Among them, F e is the end-operating force of the robot arm, j τi is the i-th column element of the robot force transfer matrix, τ i For joint J i is the moment of force, and n is the degree of freedom.

[0106] According to the above method provided by the embodiment of the present invention, a three-degree-of-freedom planar robot with the third joint under-actuated is taken as an example to illustrate the calculation process.

[0107] The schematic diagram of the three-degree-of-freedom planar manipulator structure with the third joint underactuated is as follows Figure 5 As shown. Considering the robot arm as a whole, the reaction force of the operating object on the end effector and the reaction force of the external environment on the first joint are both external forces. At this time, the second and third joints are internal mechanisms, and their force conditions do not need to be considered. In order to maintain the balance of the system, the forces acting on the first joint and the end effector should be balanced.

[0108] According to the geometric constraints, we can have:

[0109]

[0110] The end position is obtained by the following relationship:

[0111]

[0112] Based on the above formula, we can get:

[0113]

[0114] In addition, there are:

[0115]

[0116] Based on the force balance theorem:

[0117] τ1=f e d

[0118] Based on the above formula, the terminal operating force can be obtained as follows:

[0119] f e =τ1 / d

[0120] After determining the end operating force, the torque of the second joint can be solved. Next, the part of the mechanism from the second joint to the end effector is considered as a whole. The reaction force of the operating object on the end effector and the reaction force of the first link on the second joint are both external forces. In order to maintain the balance of the system, the forces acting on the second joint and the end effector should be balanced.

[0121] Therefore, we have:

[0122] τ2=f e l2sin(q3)

[0123] From the above analysis, we can know that the first joint torque can be used as an independent variable to solve the end operation force. In this case, the second joint torque becomes the dependent variable. That is, only one of the torques of the first joint and the second joint can be used as an independent variable. Otherwise, the force of the system will be unbalanced.

[0124] Substituting the obtained τ1, τ2 (τ3 = 0) into the terminal operation force calculation formula, the terminal operation force can be obtained. It can be proved that the terminal operation force calculated by the terminal operation force calculation formula is consistent with the result calculated by the above calculation formula.

[0125] The method provided by the embodiment of the present invention is used to analyze the operating force at the end of an under-actuated multi-degree-of-freedom robotic arm.

[0126] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0127] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0128] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0129] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0130] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end, characterized in that: The following steps are involved: According to the mathematical model of the underactuated multi-degree-of-freedom manipulator and the two-force balance condition, the constraint conditions of the manipulator end operating force component and torque component are established; According to the constraints of the end-operating force component and torque component of the manipulator, combined with the spatial position relationship between the manipulator mechanism and the end-operating force vector, the components of the end-operating force of the manipulator are solved; According to the components of the operating force at the end of the robot arm, the operating force at the end of the robot arm is calculated.

2. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 1, characterized in that: The constraints of the operating force components at the end of the robot arm are: The force component of the end-operation force lies in the plane formed by the underactuated joint axis and the end point.

3. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 1, characterized in that: The constraint condition of the torque component at the end of the manipulator is: The torque components lie in the plane that passes through the end points and is perpendicular to the joint axis.

4. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 1, characterized in that: According to the constraints of the end-operating force component and torque component of the manipulator, combined with the spatial position relationship between the manipulator mechanism and the end-operating force vector, the components of the end-operating force of the manipulator are solved, including: According to the spatial position of the end of the robot and the center of mass position of the underactuated joint, the constraint relationship of each element in the end-operation force component of the robot is established based on the constraint conditions of the end-operation force component of the robot; According to the direction of the underactuated joint axis, the constraint relationship of each element in the torque component of the end of the manipulator is established based on the constraint condition of the torque component of the end of the manipulator; According to the constraint relationship of each element in the end-operating force component of the robot arm and the constraint relationship of each element in the end-torque component of the robot arm, the components of the end-operating force of the robot arm are solved.

5. The method for analyzing the operating force of an underactuated multi-degree-of-freedom robot end according to claim 4, characterized in that: The constraint relationship of each element in the end-operating force component of the robot arm is: ( e p ky e z kz - e p kz e z ky ) e f ex +( e p kz e z kx - e p kx e z kz ) e f ey +( e p kx e z ky - e p ky e z kx ) e f ez =0, where e p kx , e p ky , e p kz is the direction vector of the line formed by the end effector and the center of mass of the underactuated joint, e z kx , e z ky , e z kz is the direction vector of the line formed by the joint axis and the mass center of the underactuated joint, e f ex , e f ey , e f ez is the operating force component at the end of the robot arm.

6. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 4, characterized in that: The constraint relationship of each element in the torque component of the end of the robot arm is: e z kx e n ex + e z ky e n ey + e z kz e n ez =0 in, e z kx , e z ky , e z kz is the direction vector of the line formed by the joint axis and the mass center of the underactuated joint, e n ex , e n ey , e n ez is the torque component at the end of the robot arm.

7. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 1, characterized in that: According to the components of the end-operating force of the robot arm, the end-operating force of the robot arm is calculated, including: According to the force balance relationship of the underactuated multi-DOF manipulator, the relationship between the joint force and the operating force on the end effector is established; Calculate the torque of each joint based on the relationship between the force applied to the joint and the operating force on the end effector; The end-of-arm operating force is calculated based on the torque of each joint.

8. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 7, characterized in that: The relationship between the joint force and the operating force on the end effector is: in, i f i and i f i+1 They act on joint J i and joints J i+1 The force in the coordinate system Σ i In the representation; i n i and i n i+1 They act on joint J i and joints J i+1 The moment of force in the coordinate system Σ i In the representation; i P i+1 is the coordinate system Σ i+1 The origin of the coordinate system Σ i In the representation; The joint is defined by the coordinate system ∑ i+1 To i The posture transformation matrix.

9. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 7, characterized in that: The calculation method of each joint torque is: Among them, τ i For joint J i The torque, i n i To act on joint J i The moment of force in the coordinate system Σ i In the expression, i z i For joint J i The joint axis is in the coordinate system Σ i In the representation, T is the matrix transpose.

10. The method for analyzing the operating force of an underactuated multi-degree-of-freedom manipulator end according to claim 7, characterized in that: The calculation method of the operating force at the end of the robot arm is: Among them, F e is the end-operating force of the robot arm, j τi is the i-th column element of the robot force transfer matrix, τ i For joint J i is the moment of force, and n is the degree of freedom.

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