A flexible joint space robot fast impedance control method based on contact torque compensation

Abstract

The application discloses a flexible joint space robot fast impedance control method based on contact torque compensation, first determines a state space equation of the flexible joint space robot based on a dynamic model of the flexible joint space robot, then sequentially constructs a contact torque compensator, an expected impedance model and a fixed time disturbance observer, determines compensated contact torque, impedance error intermediate value and external disturbance, then constructs a finite time impedance controller according to a singularity avoidance auxiliary function and auxiliary system state quantity, determines actual input torque of the space robot, and completes impedance control on the flexible joint space robot. The technical scheme of the application can improve the robustness of the control system, can solve the input saturation problem that may occur in the control process, can make the impedance error quickly converge, can effectively overcome the influence of external disturbance and input saturation, and improves the control precision of the contact force.

Description

Technical Field

[0001] The present invention belongs to the field of compliant control of space robots, and in particular relates to a fast impedance control method of a flexible joint space robot based on contact torque compensation. Background Art

[0002] On-orbit space services, such as on-orbit refueling, on-orbit component maintenance or replacement, space debris removal, and failed satellite recovery, are crucial for improving spacecraft operational reliability, extending spacecraft lifespan, and reducing the risk of space collisions. As an on-orbit servicing spacecraft, the flexible articulated space robot, powered by a liquid-filled flexible satellite base and equipped with a flexible articulated robotic arm, is capable of better completing various on-orbit servicing tasks and is currently a key research topic in the aerospace field.

[0003] However, contact force is generated when the end effector of a flexible joint space robot contacts the target. Excessive contact force can damage components of the space robot system and the target, seriously threatening the safety of the space robot system. Excessive contact force can make the connection between the target and the robotic arm unstable, making it easy for the target to escape the grasping control of the space robot, thus leading to mission failure. In addition, some on-orbit manufacturing and assembly service tasks require precise control of the contact force between the space robot and the target, and position control alone cannot guarantee the high control accuracy required. Therefore, how to improve the compliance and accuracy of force control during the on-orbit service of flexible joint space robots is a difficult problem that needs to be solved urgently.

[0004] Existing research results show that impedance control is an effective way to achieve compliant control of space robots, and impedance control based on dynamic models can make robots have higher response speed, accuracy and stability. Although impedance control based on dynamic models can already make robots have good environmental compliance, it does not consider the convergence speed of the controller, making it difficult to further reduce the impact of contact force. Moreover, when the desired contact force is not zero, there is a steady-state error in force control, which is not suitable for assembly operations with precise contact force requirements. In addition, instantaneous contact force may cause the control command to be too large, exceeding the maximum output of the actuator, causing input saturation and leading to system instability. How to achieve rapid impedance control of flexible joint space robots under input saturation and reduce the steady-state error of the desired contact force is a topic worthy of further research. Summary of the Invention

[0005] In view of the above problems, the object of the present invention is to provide a fast impedance control method for a flexible joint space robot based on contact torque compensation with fast convergence speed and small steady-state error.

[0006] The specific technical solutions for achieving the purpose of the present invention are as follows:

[0007] A fast impedance control method for a flexible joint space robot based on contact torque compensation includes the following steps:

[0008] Step 1: Construct a dynamic model of the flexible joint space robot and determine its state space equation;

[0009] Step 2: Construct a contact torque compensator in the joint space;

[0010] Step 3: Construct an expected impedance model in the joint space of the flexible joint space robot to characterize the impedance error;

[0011] Step 4: Construct a fixed-time disturbance observer to obtain the estimated value of the external disturbance affecting the space robot;

[0012] Step 5: Construct an auxiliary function to avoid singularity and deal with the singularity problem of virtual control law derivation in the dynamic surface method;

[0013] Step 6: Construct an anti-saturation auxiliary system to obtain auxiliary system state quantities of the processing base and joint actuators of the space robot, namely, the control torque gyro and joint motor input saturation;

[0014] Step 7. Based on the dynamic surface method and the state space equation in step 1, a finite-time impedance controller is constructed according to the interference estimation value in step 4, the singularity avoidance auxiliary function in step 5, and the auxiliary system state quantity in step 6 to complete the impedance control of the flexible joint space robot.

[0015] Compared with the prior art, the present invention has the following beneficial effects:

[0016] (1) The impedance control method constructed by the technical solution of the present invention can avoid the occurrence of singular problems based on the auxiliary function of avoiding singularity and can make the impedance error converge in a finite time, thereby improving the response speed of the control system;

[0017] (2) The fixed-time disturbance observer and the anti-saturation auxiliary system designed in the technical solution of the present invention can improve the robustness of the control system and solve the input saturation problem that may occur during the control process;

[0018] (3) The contact torque compensator in the technical solution of the present invention can greatly reduce the steady-state error of the desired contact force. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 This is a flow chart of the steps of the method for rapid impedance control of a flexible joint space robot based on contact torque compensation of the present invention.

[0020] Figure 2 This is a block diagram of the fast impedance control of the flexible joint space robot based on contact torque compensation of the present invention. DETAILED DESCRIPTION

[0021] A fast impedance control method for a flexible joint space robot based on contact torque compensation comprises the following steps:

[0022] Step 1: Construct the dynamic model of the flexible joint space robot and determine its state space equation, specifically:

[0023] Step 1-1: Construct the dynamic model of the flexible joint space robot:

[0024]

[0025]

[0026] Where, q=[q0 q m T ] T ∈R 3 (q m =[q1 q2] T ) is the rotation angle between the space robot base and the robotic arm link, θ∈R 2 is the motor angle of the space robot's manipulator joint, M(q)∈R 3×3 , J m ∈R 2×2 are the inertia matrices of the space robot and the robotic arm joint motor, is the matrix of the Coriolis force and centripetal force of the space robot, K = diag (k1, k2) is the diagonal matrix of the manipulator joint stiffness, τ0∈R 1 , τ m =[τ1 τ2] T is the control torque output by the base and the robot arm joint motor, τ d ∈R 3 is the external disturbance torque, τ e =J T (q)f e ∈R 3 is the contact torque between the space robot and the service target, J T (q) is the Jacobian matrix from the joint space to the task space of the space robot, f e is the contact force between the space robot and the service target;

[0027] Step 1-2: Determine the state space equation of the flexible joint space robot:

[0028]

[0029]

[0030] Among them, u d =M-1 (x1)τ d , K b = diag (1, k1, k2), a1 = diag (0, 1, 1), a2 = [0 I2] T , I2 is the second-order identity matrix, a3=[1 0 0] T .

[0031] The following assumptions are made:

[0032] (1) State quantity x i (i=1, 2, 3, 4) can be obtained by measurement;

[0033] (2) External interference torque τ d The derivative of is bounded, and because the inertia matrix of the space robot is bounded, the augmented error u d The derivative is bounded, that is c is an unknown positive constant;

[0034] (3) Expected trajectory Known, and Where U0 is a positive constant.

[0035] Step 2: Construct the contact torque compensator in the joint space, specifically:

[0036]

[0037] The expected contact torque after compensation is:

[0038]

[0039] Where v is the velocity factor of the contact torque compensator, t c The first contact time between the space robot and the service target, K d =diag(K d1 , K d2 , K d3 )∈R 3×3 is the desired stiffness matrix, τ f =J T (q)f d is the expected contact torque, f d It represents the contact force that the end effector is expected to exert on the service target, that is, the expected contact force.

[0040] After the expected contact force is compensated by the compensator, the steady-state error of the expected contact force of the step signal can be made 0, and the steady-state error of the expected contact force with the ramp and sinusoidal signals can be reduced.

[0041] Step 3: Construct the expected impedance model in the joint space of the flexible joint space robot and determine the intermediate impedance error vector z, specifically:

[0042] The expected impedance model is:

[0043]

[0044] The base and joint angle errors are defined as:

[0045] e1=x1-y d

[0046] The auxiliary transformation matrix is ​​as follows:

[0047]

[0048]

[0049]

[0050] The above formula can be realized by a low-pass filter. The input of the filter is The output is τ a , expected impedance model parameter M d , C d , K d The selection needs to meet the following requirements: Thus, the existence of auxiliary transformation matrices Λ, Γ is guaranteed;

[0051] The augmented impedance error It can be expressed as:

[0052]

[0053]

[0054] Among them, Λ, Γ are both positive definite diagonal matrices, M d =diag(M d1 , M d2 , M d3 ), C d =diag(C d1 , C d2 , C d3 ) represent the desired inertia matrix and the desired damping matrix, represents the expected contact torque after compensation, τ a is the intermediate variable, y d represents the expected trajectory.

[0055] It can be found that and are equivalent, the impedance error intermediate vector z can be regarded as the output of the first-order low-pass filter, and the input is the augmented impedance error Therefore, in the low frequency range, ||z||→0 means At this time, there is ||w e ||→0, that is, the system impedance error tends to zero, achieving the goal of impedance control.

[0056] Step 4: Construct a fixed-time disturbance observer to obtain the estimated value of the external disturbance affecting the space robot. Specifically:

[0057]

[0058] Among them, for x=[x1 x2 ... x n ] T ∈R n and α≥0, define |x| α =[|x1| α |x2| α ... |x n | α ] T sig α (x)=[|x1| α sign(x1) |x2| α sign(x2) ... |x n | α sign(x n )] T , sign(·) represents the sign function;

[0059] and is the disturbance observer state vector, representing the system state x2 and external disturbance u d The estimated value of g1>0, g2>0, α∈(1,1.5) and β∈(0.5,1) are the fixed time disturbance observer coefficients, and satisfy δ∈(0, 1) is the amplification factor.

[0060] Step 5: Construct an auxiliary function to avoid singularity and deal with the singularity problem of the derivative of the virtual control law in the dynamic surface method. Specifically:

[0061] sig α (x)=[|x1| α sign(x1) |x2| α sign(x2) ... |x n | α sign(x n )] T

[0062]

[0063] Among them, ε a is a small positive constant.

[0064] Step 6: Build an anti-saturation auxiliary system to obtain the auxiliary system state quantities of the processing base and joint actuator of the space robot, namely the control torque gyro and joint motor input saturation, specifically:

[0065]

[0066]

[0067] h1=diag(h 11 , h 12 , h 13 )

[0068] h2=diag(h 21 , h 22 , h 23 )

[0069] h3=diag(h 31 , h 32 )

[0070] h4=diag(h 41 , h 42 )

[0071] g b (χ b )=[g(χ b1 ) g(χ b2 ) g(χ b3 )] T

[0072] Δτ0=τ0-τ c0

[0073] Δτ m =τ m -τ cm

[0074] Among them, χ b and χ m They are the auxiliary system state quantities for processing the base and joint actuator, namely the control torque gyro and joint motor input saturation, respectively. h1, h2, h3 and h4 are the auxiliary system parameters.

[0075] Step 7: Based on the dynamic surface method and the state space equation in step 1, according to the interference estimation value in step 4, the singularity avoidance auxiliary function in step 5, and the auxiliary system state quantity in step 6, a finite-time impedance controller is constructed to complete the impedance control of the flexible joint space robot. Specifically:

[0076] Step 7-1: Determine the virtual control law x of the flexible joint space robot 3，d And the actual control law of the base τ c0 :

[0077] x 3，d =a2τ cd

[0078]

[0079]

[0080] Considering the influence of input saturation, the transition state s is defined b and s m as follows:

[0081]

[0082] The error surface is defined as follows:

[0083]

[0084] Where: ω1, ω2∈R 2 For a nonlinear filter:

[0085] The output value of

[0086] Where: σ j is the time constant, x i+2，d is the virtual control law;

[0087] The reference velocity and reference acceleration of the base and joint angle are defined as:

[0088]

[0089] in, is the controller parameter, g1(s b )=[g(s b1 ) g(s b2 ) g(s b3 )] T ;

[0090] Step 7-2: Determine the virtual control law x of the flexible joint space robot 4，d :

[0091]

[0092]

[0093] in:

[0094] g2(s2)=[g(s 21 ) g(s 22 )] T ,

[0095] Step 7-3: Determine the actual control law τ of the flexible joint space robot arm cm :

[0096]

[0097] in, is the controller parameter;

[0098] Step 7-4: Actual control law τ for the flexible joint space robot base and manipulator c0 and τ cm Apply physical constraints to obtain the actual input torque of the flexible joint space robot base and manipulator:

[0099]

[0100] Among them, τ imax and τ imin Respectively represent the maximum torque in the forward and reverse directions that the actuator can provide.

[0101] The present invention will be further described below with reference to the embodiments.

[0102] Example

[0103] Combine Figure 1 and Figure 2 , a fast impedance control method for a flexible joint space robot based on contact torque compensation, comprising the following steps:

[0104] Step 1: Construct the dynamic model of the flexible joint space robot and determine its state space equation, specifically:

[0105] Step 1-1: Construct the dynamic model of the flexible joint space robot:

[0106]

[0107]

[0108] Where, q=[q0 q m T ] T ∈R3 (q m =[q1 q2] T ) is the rotation angle between the space robot base and the robotic arm link, θ∈R 2 is the motor angle of the space robot's manipulator joint, M(q)∈R 3×3 , J m ∈R 2×2 are the inertia matrices of the space robot and the robotic arm joint motor, is the matrix of the Coriolis force and centripetal force of the space robot, K = diag (k1, k2) is the diagonal matrix of the manipulator joint stiffness, τ0∈R 1 , τ m =[τ1 τ2] T is the control torque output by the base and the robot arm joint motor, τ d ∈R 3 is the external disturbance torque, τ e =J T (q)f e ∈R 3 is the contact torque between the space robot and the service target, J T (q) is the Jacobian matrix from the joint space to the task space of the space robot, f e is the contact force between the space robot and the service target;

[0109] Step 1-2: Determine the state space equation of the flexible joint space robot:

[0110]

[0111]

[0112] Among them, u d =M -1 (x1)τ d , K b = diag (1, k1, k2), a1 = diag (0, 1, 1), a2 = [0 I2] T , I2 is the second-order identity matrix, a3=[1 0 0] T .

[0113] The following assumptions are made:

[0114] (1) State quantity x i (i=1, 2, 3, 4) can be obtained by measurement;

[0115] (2) External interference torque τ d The derivative of is bounded, and because the inertia matrix of the space robot is bounded, the augmented error u d The derivative is bounded, that is c is an unknown positive constant;

[0116] (3) Expected trajectory Known, and Where U0 is a positive constant.

[0117] Step 2: Construct the contact torque compensator in the joint space, specifically:

[0118]

[0119] The expected contact torque after compensation is:

[0120]

[0121] Where v is the velocity factor of the contact torque compensator, t c The first contact time between the space robot and the service target, K d =diag(K d1 , K d2 , K d3 )∈R 3×3 is the desired stiffness matrix, τ f =J T (q)f d is the expected contact torque, f d It represents the contact force that the end effector is expected to exert on the service target, that is, the expected contact force.

[0122] After the expected contact force is compensated by the compensator, the steady-state error of the expected contact force of the step signal can be made 0, and the steady-state error of the expected contact force with the ramp and sinusoidal signals can be reduced.

[0123] Step 3: Construct the expected impedance model in the joint space of the flexible joint space robot to characterize the impedance error, specifically:

[0124] The expected impedance model is:

[0125]

[0126] The base and joint angle errors are defined as:

[0127] e1=x1-y d

[0128] The auxiliary transformation matrix is ​​as follows:

[0129]

[0130]

[0131]

[0132] The above formula can be realized by a low-pass filter. The input of the filter is The output is τ a , expected impedance model parameter M d , C d , K d The selection needs to meet the following requirements: Thus, the existence of auxiliary transformation matrices Λ, Γ is guaranteed;

[0133] The augmented impedance error It can be expressed as:

[0134]

[0135]

[0136] Among them, Λ, Γ are both positive definite diagonal matrices, M d =diag(M d1 , M d2 , M d3 ), C d =diag(C d1 , C d2 , C d3 ) represent the desired inertia matrix and the desired damping matrix, represents the expected contact torque after compensation, τ a is the intermediate variable, y d represents the expected trajectory.

[0137] It can be found that with ||w e ||→0 is equivalent, the impedance error intermediate vector z can be regarded as the output of the first-order low-pass filter, and the input is the augmented impedance error Therefore, in the low frequency range, ||z||→0 means At this time, there is ||w e ||→0, that is, the system impedance error tends to zero, achieving the goal of impedance control.

[0138] Step 4: Construct a fixed-time disturbance observer to obtain the estimated value of the external disturbance affecting the space robot. Specifically:

[0139]

[0140] Among them, for x=[x1 x2 ... x n ] T ∈R n and α≥0, define |x| α =[|x1| α |x2| α ... |x n |α ] T sig α (x)=[|x1| α sign(x1) |x2| α sign(x2) ... |x n | a sign(x n )] T , sign(·) represents the sign function;

[0141] and is the disturbance observer state vector, representing the system state x2 and external disturbance u d The estimated value of g1>0, g2>0, α∈(1,1.5) and β∈(0.5,1) are the fixed time disturbance observer coefficients, and satisfy δ∈(0, 1) is the amplification factor.

[0142] Step 5: Construct an auxiliary function to avoid singularity and deal with the singularity problem of the derivative of the virtual control law in the dynamic surface method. Specifically:

[0143] sig α (x)=[|x1| α sign(x1) |x2| α sign(x2) ... |x n | α sign(x n )] T

[0144]

[0145] Among them, ε a is a small positive constant.

[0146] Step 6: Build an anti-saturation auxiliary system to obtain the auxiliary system state quantities of the processing base and joint actuator of the space robot, namely the control torque gyro and joint motor input saturation, specifically:

[0147]

[0148]

[0149] h1=diag(h 11 , h 12 , h 13 )

[0150] h2=diag(h 21 , h 22 , h 23 )

[0151] h3=diag(h 31 , h 32 )

[0152] h4=diag(h 41 , h 42 )

[0153] g b (χ b )=[g(χ b1 ) g(χ b2 ) g(χ b3 )] T

[0154] Δv0=τ0-τ c0

[0155] Δτ m =τ m -τ cm

[0156] Among them, χ b and χ m They are the auxiliary system state quantities for processing the base and joint actuator, namely the control torque gyro and joint motor input saturation, respectively. h1, h2, h3 and h4 are the auxiliary system parameters.

[0157] Step 7: Based on the dynamic surface method and the state space equation in step 1, according to the interference estimation value in step 4, the singularity avoidance auxiliary function in step 5, and the auxiliary system state quantity in step 6, a finite-time impedance controller is constructed to complete the impedance control of the flexible joint space robot. Specifically:

[0158] Step 7-1: Determine the virtual control law x of the flexible joint space robot 3，d And the actual control law of the base τ c0 :

[0159] x 3，d =a2τ cd

[0160]

[0161]

[0162] Considering the influence of input saturation, the transition state s is defined b and s m as follows:

[0163]

[0164] The error surface is defined as follows:

[0165]

[0166] Where: ω1, ω2∈R 2 For a nonlinear filter:

[0167] ω i (0) = x i+2，d (0), output value of i=1,2;

[0168] Where: σ j is the time constant, x i+2，d is the virtual control law;

[0169] The reference velocity and reference acceleration of the base and joint angle are defined as:

[0170]

[0171] in, is the controller parameter, g1(s b )=[g(s b1 ) g(s b2 ) g(s b3 )] T ;

[0172] Step 7-2: Determine the virtual control law x of the flexible joint space robot 4，d :

[0173]

[0174]

[0175] in:

[0176] g2(s2)=[g(s 21 ) g(s 22 )] T ,

[0177] Step 7-3: Determine the actual control law τ of the flexible joint space robot arm cm :

[0178]

[0179] in, is the controller parameter;

[0180] Step 7-4: Actual control law τ for the flexible joint space robot base and manipulator c0 and τ cm Apply physical constraints to obtain the actual input torque of the flexible joint space robot base and manipulator:

[0181]

[0182] Among them, τ imax and τ imin Respectively represent the maximum torque in the forward and reverse directions that the actuator can provide.

[0183] The fast impedance control method of the flexible joint space robot based on contact torque compensation proposed in the present invention can quickly converge the impedance error, effectively overcome the influence of external disturbances and input saturation, and improve the control accuracy of the contact force.

[0184] The above embodiments illustrate and describe the basic principles and main features of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention, and such changes and modifications fall within the scope of the invention as claimed.

Claims

1. A fast impedance control method for a flexible joint space robot based on contact torque compensation, characterized in that: The following steps are involved: Step 1: Construct the dynamic model of the flexible joint space robot and determine its state space equation: Step 1-1: Construct the dynamic model of the flexible joint space robot: Where, q=[q0 q m T ] T ∈R 3 (q m =[q1 q2] T ) is the rotation angle between the space robot base and the robotic arm link, θ∈R 2 is the motor angle of the space robot's manipulator joint, M(q)∈R 3×3 ,J m ∈R 2×2 are the inertia matrices of the space robot and the robotic arm joint motor, is the matrix of the Coriolis force and centripetal force of the space robot, K = diag (k1, k2) is the diagonal matrix of the joint stiffness of the manipulator, τ0∈R 1 ,τ m =[τ1 τ2] T is the control torque output by the base and the robot arm joint motor, τ d ∈R 3 is the external disturbance torque, τ e =J T (q)f e ∈R 3 is the contact torque between the space robot and the service target, J T (q) is the Jacobian matrix from the joint space to the task space of the space robot, f e is the contact force between the space robot and the service target; Step 1-2: Determine the state space equation of the flexible joint space robot: Among them, u d =M -1 (x1)τ d , K b =diag(1,k1,k2),a1=diag(0,1,1),a2=[0I2] T , I2 is the 2nd-order identity matrix, a3=[100] T ; Step 2: Construct the contact torque compensator in the joint space: Where v is the velocity factor of the contact torque compensator, t c The first contact time between the space robot and the service target, K d =diag(K d1 ,K d2 ,K d3 )∈R 3×3 is the desired stiffness matrix, τ f =J T (q)f d is the expected contact torque, f d represents the contact force that the end effector is expected to exert on the service target, that is, the expected contact force; Step 3: Construct the expected impedance model in the joint space of the flexible joint space robot, characterize the impedance error, and determine the impedance error intermediate vector z: e1=x1-y d Among them, Λ, Γ are both positive definite diagonal matrices, M d =diag(M d1 ,M d2 ,M d3 ) represents the desired inertia matrix and the desired damping matrix, represents the expected contact torque after compensation, τ a is the intermediate variable, y d represents the expected trajectory; Step 4: Construct a fixed-time disturbance observer to obtain the estimated value of the external disturbance affecting the space robot: in, and is the disturbance observer state vector, representing the system state x2 and external disturbance u d The estimated value of g1>0, g2>0, α∈(1,1.5) and β∈(0.5,1) are the fixed time disturbance observer coefficients, and satisfy δ∈(0,1) is the amplification factor; Step 5: Construct an auxiliary function to avoid singularities and handle the singularity problem of virtual control law derivation in the dynamic surface method: Among them, ε a is a very small positive constant; Step 6: Build an anti-saturation auxiliary system to obtain the auxiliary system state quantities of the processing base and joint actuator of the space robot, namely the control torque gyro and joint motor input saturation: h1=diag(h 11 ,h 12 ,h 13 ) h2=diag(h 21 ,h 22 ,h 23 ) h3=diag(h 31 ,h 32 ) h4=diag(h 41 ,h 42 ) g b (x b )=[g(x b1 ) g(x b2 ) g(x b3 )] T Δτ0=τ0-τ c0 Dt m =t m -t cm Among them, χ b and χ m They are the auxiliary system state quantities for processing the base and joint actuator, i.e., the control torque gyro and joint motor input saturation, respectively. h1, h2, h3, and h4 are the auxiliary system parameters. Step 7. Based on the dynamic surface method and the state space equation in step 1, according to the interference estimation value in step 4, the auxiliary function for avoiding singularities in step 5, and the auxiliary system state quantity in step 6, a finite-time impedance controller is constructed to determine the actual input torque of the space robot, thus completing the impedance control of the flexible joint space robot: Step 7-1: Determine the virtual control law x of the flexible joint space robot 3,d And the actual control law of the base τ c0 : x 3,d =a2τ cd in, is the controller parameter, g1(s b )=[g(s b1 ) g(s b2 ) g(s b3 )] T ; s i represents the error surface, s b and s m Represents the transition state, ω1,ω2∈R 2 is a nonlinear filter, represents the reference angular velocity, represents the reference angular acceleration; Step 7-2: Determine the virtual control law x of the flexible joint space robot 4,d : Where: i is the time constant; Step 7-3: Determine the actual control law τ of the flexible joint space robot arm cm : in, is the controller parameter; Step 7-4: Actual control law τ for the flexible joint space robot base and manipulator c0 and τ cm Apply physical constraints to obtain the actual input torque of the flexible joint space robot base and manipulator: Among them, τ imax and τ imin Respectively represent the maximum torque in the forward and reverse directions that the actuator can provide.

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