CN114211492B - Optimal track planning method of multi-degree-of-freedom mechanical arm based on model - Google Patents
Optimal track planning method of multi-degree-of-freedom mechanical arm based on model Download PDFInfo
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Abstract
The invention provides a model-based optimal trajectory planning method of a multi-degree-of-freedom mechanical arm, which is completed by matching an upper computer, a mechanical arm controller, a feedback control module, a mechanical arm and an optimal trajectory planning module.
Description
Technical Field
The invention belongs to the technical field of program control of mechanical arm operation, and particularly relates to an optimal track planning method of a multi-degree-of-freedom mechanical arm based on a model.
Background
The mechanical arm has the characteristics of flexibility, strong expansibility and the like, is rapidly developed under the large background of industrial automation, and is widely applied to industrial production, medical assistance and space exploration. The motion control of the mechanical arm only considers kinematics, and the basic motion control of the mechanical arm can be realized. When the motion speed of the mechanical arm is required to be high, only the control method of the kinematics of the mechanical arm is considered, and the ideal moment of each joint of the mechanical arm can not be ensured to be always in the rated moment range, so that the mechanical arm controller can not give the ideal moment, and the track tracking error is caused. Therefore, in order to meet the high-speed and high-precision requirements of the movement of the mechanical arm, the dynamics system of the mechanical arm needs to be considered, and the two points of the conventional optimal trajectory planning method of the mechanical arm are not enough: firstly, compared with a track planning method with a control law as a result, the existing track planning method with a track as a result for four-axis and above mechanical arms relies on a feedback control system of the mechanical arms, and track tracking precision is difficult to ensure under a scene of high-speed movement of the mechanical arms; secondly, the inertial tensor matrix analysis expression of the four-axis and above mechanical arm is long, the inverse analysis derivation is more complex, the method is difficult to apply to a program algorithm, and the optimal control method cannot be applied to the optimal track planning problem of the mechanical arm, so that the optimal track planning method of the multi-degree-of-freedom mechanical arm based on the model is provided.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides an optimal trajectory planning method of a multi-degree-of-freedom mechanical arm based on a model, which is reasonable in design, and establishes a mechanical arm dynamics system model by adopting a non-causal modeling language, so that differential state equations of four-axis and above mechanical arm systems are expressed, and the application of an optimal control method in the task of planning the optimal trajectory of the four-axis and above mechanical arms can be realized.
In order to achieve the above object, the present invention is realized by the following technical scheme: the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm based on the model is completed by matching an upper computer, a mechanical arm controller, a feedback control module, the mechanical arm and an optimal trajectory planning module, and the functions of all the parts are as follows:
(1) The upper computer: the teaching function mainly comprises a related function of general track planning, namely, a function of controlling the mechanical arm through controlling the rotation angle of each joint of a joint space or controlling the mechanical arm to reach the set initial posture after performing inverse kinematics calculation;
(2) And the mechanical arm controller is used for: the module drives each joint motor of the mechanical arm to move according to an ideal track by receiving a feedforward control instruction output by a control law method in the optimal track planning module or receiving a feedback control instruction in a feedback control module;
(3) And a feedback control module: the module receives general track information issued by an upper computer or optimal track information output by a track method in an optimal track planning module, and receives track information acquired from a controlled mechanical arm object through a sensor to perform feedback control, and an output feedback control instruction directly acts on a mechanical arm controller;
(5) Mechanical arm: the module is used as a controlled object, is controlled by the mechanical arm controller to move, is provided with a related sensor, and can transmit the movement information of the mechanical arm to the feedback control module;
(4) And an optimal track planning module: according to the initial posture and the target posture of the mechanical arm given in the upper computer, the module combines the general description of the optimization objective construction optimization problem, including selection of performance indexes, establishment of inequality constraint, setting of boundary conditions and establishment of differential state equation constraint of the mechanical arm system based on a model, finally, the problem is solved by using an optimal control method, and a solving result is input into a mechanical arm controller as feedforward of the mechanical arm, wherein each part of the optimization problem is described as follows:
(1) performance index: the method comprises four indexes of minimum time, minimum energy consumption, minimum impact and optimal mixing;
(2) inequality constraint: the maximum rotation angle constraint of each joint of the mechanical arm, the maximum rotation speed constraint, the maximum rotation acceleration constraint and the maximum output torque/current constraint of each joint motor are included;
(3) boundary conditions: the method comprises the steps of setting rotation angle information of each joint of the mechanical arm in a joint space corresponding to an initial time and a termination time in an upper computer, and angular velocity and angular acceleration information of each joint of the mechanical arm corresponding to the time;
(4) differential state equation: based on the model method, the expression of the differential state equation of the mechanical arm is simplified by using a black box function, so that the application of the optimal control method to the optimal track planning problem of four-axis and above mechanical arms is realized, and the input and output of the differential state equation are u and x respectively 0 ,t,x,
y, the meanings represented by the variables and the corresponding relation with the mechanical arm system are as follows:
(1) u—input of differential state equation black box function, corresponding to output torque/current (τ/i) of each joint motor in the mechanical arm system;
②x 0 -initial state of the differential state equation black box function, corresponding to the rotation angle value (q 0 )；
(3) t-system run time with the initial state of the system as the initial time;
(4) x-state variable of the differential state equation black box function, corresponding to the rotation angle value and rotation angular velocity value of each joint in the mechanical arm system along with time change
⑤
-differentiation of the state variables of the differential state equation black box function, corresponding to the rotational angular velocity values, rotational angular acceleration values of the joints in the robotic arm system over time +.>
(6) y—the output of the differential state equation black box function, which can be defined as any variable in the system where it is set to correspond to the torque/current (τ/i) of each joint in the robotic arm system over time;
in summary, based on the model, the kinetic equation of the mechanical arm system is written as:
the differential state equation of the robotic arm system can be written as:
the solving technical route of the optimal control method is as follows:
(1) Initializing a mechanical arm system;
the method comprises the steps of inputting an initial gesture and a target gesture of the mechanical arm, and operating the mechanical arm to reach the initial gesture;
(2) Describing an optimal control problem;
the method comprises the steps of selecting performance indexes, establishing inequality constraints, giving boundary conditions and constructing differential state equation constraints of a mechanical arm system based on a model;
(3) Discretizing the optimal control problem by using a transcription method;
the method comprises the steps of converting an optimal control problem into a nonlinear programming problem by using a point matching method and a targeting method;
(4) And solving the nonlinear programming problem by using a nonlinear programming solver.
As a preferred embodiment of the present invention, in (4) of the optimal trajectory planning module, the differential state equation refers to an expression form of a system dynamics equation in a differential state space, and for a dynamics equation of a complex system, the expression form of the differential state space is difficult to be analytically derived, so a model-based method is used.
In the (1) of the optimal trajectory planning module, the time is minimum, the energy consumption is minimum, the impact is minimum, and the mixing is maximum, wherein the first three indexes respectively correspond to the improvement of the working efficiency of the mechanical arm, the reduction of the energy consumption of the mechanical arm, and the inhibition of the vibration impact caused by the abrupt change of the control quantity of the mechanical arm, and the mixing is optimal, namely, the motion performance of the mechanical arm is optimized by comprehensively considering the effects of the indexes.
In the preferred embodiment of the present invention, in the (1) of the optimum trajectory planning module, the mixed optimum four types of indexes are any two or more types of indexes of minimum time, minimum energy consumption and minimum impact.
The invention has the beneficial effects that:
1. the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm based on the model adopts a non-causal modeling language to establish a mechanical arm dynamics system model, so that differential state equations of four-axis and above mechanical arm systems are expressed, and the application of the optimal control method in the optimal trajectory planning task of the four-axis and above mechanical arm can be realized.
2. According to the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm based on the model, an optimal control method is adopted to solve the optimal trajectory planning problem of the mechanical arm with four or more axes, the input control law required by the mechanical arm controller is directly obtained to perform feedforward control, dependence of the mechanical arm on feedback control is effectively reduced, and the trajectory tracking precision of the mechanical arm under a high-speed motion scene is improved.
Drawings
FIG. 1 is a schematic flow chart of an optimal planning problem of a solving mechanical arm by an optimal control method of an optimal trajectory planning method of a multi-degree-of-freedom mechanical arm based on a model;
FIG. 2 is a flow chart of the optimal trajectory planning of the mechanical arm of the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm based on the model of the invention;
FIG. 3 is a diagram of a differential state equation based on a model of the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm according to the present invention;
fig. 4 is a partial schematic diagram of a structure diagram of a model-based differential state equation of the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm according to the present invention.
Detailed Description
The invention is further described in connection with the following detailed description, in order to make the technical means, the creation characteristics, the achievement of the purpose and the effect of the invention easy to understand.
Referring to fig. 1 to 4, the present invention provides a technical solution: the optimal trajectory planning method of the multi-degree-of-freedom mechanical arm based on the model is completed by matching an upper computer, a mechanical arm controller, a feedback control module, the mechanical arm and an optimal trajectory planning module, and the functions of all the parts are as follows:
(1) The upper computer: the teaching function mainly comprises a related function of general track planning, namely, a function of controlling the mechanical arm through controlling the rotation angle of each joint of a joint space or controlling the mechanical arm to reach the set initial posture after performing inverse kinematics calculation;
(2) And the mechanical arm controller is used for: the module drives each joint motor of the mechanical arm to move according to an ideal track by receiving a feedforward control instruction output by a control law method in the optimal track planning module or receiving a feedback control instruction in a feedback control module;
(3) And a feedback control module: the module receives general track information issued by an upper computer or optimal track information output by a track method in an optimal track planning module, and receives track information acquired from a controlled mechanical arm object through a sensor to perform feedback control, and an output feedback control instruction directly acts on a mechanical arm controller;
(5) Mechanical arm: the module is used as a controlled object, is controlled by the mechanical arm controller to move, is provided with a related sensor, and can transmit the movement information of the mechanical arm to the feedback control module;
(4) And an optimal track planning module: according to the initial posture and the target posture of the mechanical arm given in the upper computer, the module combines the general description of the optimization objective construction optimization problem, including selection of performance indexes, establishment of inequality constraint, setting of boundary conditions and establishment of differential state equation constraint of the mechanical arm system based on a model, finally, the problem is solved by using an optimal control method, and a solving result is input into a mechanical arm controller as feedforward of the mechanical arm, wherein each part of the optimization problem is described as follows:
(1) performance index: the method comprises four indexes of minimum time, minimum energy consumption, minimum impact and optimal mixing;
(2) inequality constraint: the maximum rotation angle constraint of each joint of the mechanical arm, the maximum rotation speed constraint, the maximum rotation acceleration constraint and the maximum output torque/current constraint of each joint motor are included;
(3) boundary conditions: the method comprises the steps of setting rotation angle information of each joint of the mechanical arm in a joint space corresponding to an initial time and a termination time in an upper computer, and angular velocity and angular acceleration information of each joint of the mechanical arm corresponding to the time;
(4) differential state equation: based on the model method, the expression of the differential state equation of the mechanical arm is simplified by using a black box function, so that the application of the optimal control method to the optimal track planning problem of four-axis and above mechanical arms is realized, and the input and output of the differential state equation are u and x respectively 0 ,t,x,
t, the meanings represented by the variables and the corresponding relation with the mechanical arm system are as follows:
(1) u—input of differential state equation black box function, corresponding to output torque/current (τ/i) of each joint motor in the mechanical arm system;
②x 0 -initial state of the differential state equation black box function, corresponding to the rotation angle value (q 0 )；
(3) t-system run time with the initial state of the system as the initial time;
(4) x-state variable of the differential state equation black box function, corresponding to the rotation angle value and rotation angular velocity value of each joint in the mechanical arm system along with time change
⑤
-differentiation of the state variables of the differential state equation black box function, corresponding to the rotational angular velocity values, rotational angular acceleration values of the joints in the robotic arm system over time +.>
(6) y—the output of the differential state equation black box function, which can be defined as any variable in the system where it is set to correspond to the torque/current (τ/i) of each joint in the robotic arm system over time;
in summary, based on the model, the kinetic equation of the mechanical arm system is written as:
the differential state equation of the robotic arm system can be written as:
the solving technical route of the optimal control method is as follows:
(1) Initializing a mechanical arm system;
the method comprises the steps of inputting an initial gesture and a target gesture of the mechanical arm, and operating the mechanical arm to reach the initial gesture;
(2) Describing an optimal control problem;
the method comprises the steps of selecting performance indexes, establishing inequality constraints, giving boundary conditions and constructing differential state equation constraints of a mechanical arm system based on a model;
(3) Discretizing the optimal control problem by using a transcription method;
the method comprises the steps of converting an optimal control problem into a nonlinear programming problem by using a point matching method and a targeting method;
(4) And solving the nonlinear programming problem by using a nonlinear programming solver.
As a preferred embodiment of the present invention, in (4) of the optimal trajectory planning module, the differential state equation refers to an expression form of a system dynamics equation in a differential state space, and for a dynamics equation of a complex system, the expression form of the differential state space is difficult to be analytically derived, so a model-based method is used.
In the (1) of the optimal trajectory planning module, the time is minimum, the energy consumption is minimum, the impact is minimum, and the mixing is maximum, wherein the first three indexes respectively correspond to the improvement of the working efficiency of the mechanical arm, the reduction of the energy consumption of the mechanical arm, and the inhibition of the vibration impact caused by the abrupt change of the control quantity of the mechanical arm, and the mixing is optimal, namely, the motion performance of the mechanical arm is optimized by comprehensively considering the effects of the indexes.
In the preferred embodiment of the present invention, in the (1) of the optimum trajectory planning module, the mixed optimum four types of indexes are any two or more types of indexes of minimum time, minimum energy consumption and minimum impact.
In this embodiment, an optimal trajectory planning flowchart of the four-axis and above mechanical arm is shown in fig. 2, where the execution steps of the optimal trajectory planning method with the control law result are as follows:
(1) An operator sets the target pose of the mechanical arm through the upper computer, and signals are sent to the optimal track planning module;
(2) The optimal track planning method with the control result being the optimal track is selected to directly obtain the control law of the optimal track, and the control law can be directly used as the feedforward input of the mechanical arm controller;
in fig. 2, it can be observed that when the optimum trajectory planning method whose result is trajectory is selected, the planned result is trajectory, that is, angle, angular velocity, angular acceleration information of each joint. The track signal can be transmitted to the mechanical arm controller only after being converted into a control law by a feedback control system, and the track signal is essentially controlled by the optimal track through feedback control. Therefore, in the scene of high-speed movement of the mechanical arm, the optimal track planning method with the control law is selected to realize feedforward control of the mechanical arm system, compared with the optimal track planning method with the control law as the track, the feedback control of the mechanical arm system is realized, the dependence of the mechanical arm on the feedback control system can be reduced, the track tracking precision of the mechanical arm can be ensured, and the movement performance of the mechanical arm is improved;
(3) The mechanical arm controller controls the mechanical arm to realize an optimal track;
(4) The sensor acquires the position information of the mechanical arm to perform feedback control so as to further ensure the track tracking precision;
in this embodiment, the optimal control method module implements optimal energy consumption trajectory planning of the six-axis mechanical arm, referring to fig. 3, and the execution steps are as follows:
(1) Establishing a kinematic and dynamic model of the mechanical arm;
in this embodiment, the mapping relationship between the representation of any posture of the mechanical arm in the cartesian space and the representation in the joint space is established by establishing the fixedly connected local coordinate system on each connecting rod of the mechanical arm, and the common methods include a D-H coordinate establishment method and the like.
Through the establishment of the kinematic mapping relationship, basic information is provided for the calculation of the dynamic model of the mechanical arm; in this embodiment, the kinetic equation of the mechanical arm can be deduced by combining the kinematic information of the mechanical arm, and the common methods include a Newton-Euler method and a Lagrange method; based on the method, a non-causal modeling language can be used for modeling a kinematic model of the mechanical arm, and the dynamics equation can be abbreviated as:
the model can be used as a description of the kinetic differential equation in the optimal control problem;
(2) Describing optimal control problems including performance index functions, dynamic differential equations, boundary conditions and inequality constraints;
general optimal control problems can be described as:
performance index:
wherein J represents a performance index, phi represents a terminal cost function, L represents a process cost function, t 0 Indicating the starting time, t f Indicating the termination time, x indicating the state variable, u indicating the control variable;
system differential state equation:
wherein,,
differential representing a state variable;
boundary conditions:
ψ(x(t 0 ),t 0 ,x(t f ),t f )＝0
inequality constraint:
C(x(t),u(t),t)≤0
for the problem of energy consumption optimal control of a six-axis mechanical arm, it can be described as:
performance index:
wherein τ i Representing control variables and representing control moment of each joint of the mechanical arm i;
system differential state equation:
wherein, q is,
representing state variables, namely representing the angles and angular velocities of joints of the mechanical arm; />
The differential of the state variable is represented, and the angular acceleration of each joint of the mechanical arm is represented by the differential; forward model represents a positive model of a kinetic model constructed using a non-causal modeling language;
boundary conditions:
q(0)＝(q 0 ) 6×1 ,q(t f )＝(q f ) 6×1
wherein q 0 ,
q f ,/>
The angle and the angular velocity of each joint initial state of the mechanical arm are respectively shown, and the angle and the angular velocity of the terminal state are all 6 multiplied by 1 constant vectors which can be set by a user;
inequality constraint:
q||≤q max
||τ||≤τ max
wherein q max ,
τ max The maximum values of the angles, angular velocities and driving moments of the joints of the mechanical arm are respectively represented, and are generally set as constant vectors of 6 multiplied by 1 by a user;
(3) Converting the optimal control problem into a nonlinear programming problem by using a transcription method;
the transcription method is a problem transformation framework capable of mapping the optimal control problem of a continuous state space to the nonlinear programming problem of a discrete state space, the core discrete method is called a point matching method, the point matching method is mainly divided into two major types, namely a global method and a local method, and the global method comprises an LGL pseudo-spectrum method and the like; the local method is classified into trapezoidal method, simpson method, etc., and can be used herein; the targeting rule can ensure the conversion relation between adjacent state points after dispersion, and a series of constraints established by the targeting rule correspond to differential state equations in generalized description of the optimal control problem;
(4) Solving by a nonlinear programming solver to obtain an optimal track, namely moment, angle and angular speed of each joint at each discrete time node;
and solving the problems by adopting a nonlinear programming solver, and selecting solvers such as IPOPT, SNOPT and the like to solve the nonlinear programming problem converted from the original optimal control problem.
While the fundamental and principal features of the invention and advantages of the invention have been shown and described, it will be apparent to those skilled in the art that the invention is not limited to the details of the foregoing exemplary embodiments, but may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
Furthermore, it should be understood that although the present disclosure describes embodiments, not every embodiment is provided with a separate embodiment, and that this description is provided for clarity only, and that the disclosure is not limited to the embodiments described in detail below, and that the embodiments described in the examples may be combined as appropriate to form other embodiments that will be apparent to those skilled in the art.
Claims (4)
1. The optimal trajectory planning method of the multi-degree-of-freedom mechanical arm based on the model is characterized by comprising an upper computer, a mechanical arm controller, a feedback control module, a mechanical arm and an optimal trajectory planning module, wherein the functions of the upper computer, the mechanical arm controller, the feedback control module, the mechanical arm and the optimal trajectory planning module are completed in a matching mode, and the functions of the upper computer, the mechanical arm controller, the feedback control module, the mechanical arm and the optimal trajectory planning module are as follows:
(1) The upper computer: the teaching function mainly comprises a related function of general track planning, namely, a function of controlling the mechanical arm through controlling the rotation angle of each joint of a joint space or controlling the mechanical arm to reach the set initial posture after performing inverse kinematics calculation;
(2) And the mechanical arm controller is used for: the module drives each joint motor of the mechanical arm to move according to an ideal track by receiving a feedforward control instruction output by a control law method in the optimal track planning module or receiving a feedback control instruction in a feedback control module;
(3) And a feedback control module: the module receives general track information issued by an upper computer or optimal track information output by a track method in an optimal track planning module, and receives track information acquired from a controlled mechanical arm object through a sensor to perform feedback control, and an output feedback control instruction directly acts on a mechanical arm controller;
(5) Mechanical arm: the module is used as a controlled object, is controlled by the mechanical arm controller to move, is provided with a related sensor, and can transmit the movement information of the mechanical arm to the feedback control module;
(4) And an optimal track planning module: according to the initial posture and the target posture of the mechanical arm given in the upper computer, the module combines the general description of the optimization objective construction optimization problem, including selection of performance indexes, establishment of inequality constraint, setting of boundary conditions and establishment of differential state equation constraint of the mechanical arm system based on a model, finally, the problem is solved by using an optimal control method, and a solving result is input into a mechanical arm controller as feedforward of the mechanical arm, wherein each part of the optimization problem is described as follows:
(1) performance index: the method comprises four indexes of minimum time, minimum energy consumption, minimum impact and optimal mixing;
(2) inequality constraint: the maximum rotation angle constraint of each joint of the mechanical arm, the maximum rotation speed constraint, the maximum rotation acceleration constraint and the maximum output torque/current constraint of each joint motor are included;
(3) boundary conditions: the method comprises the steps of setting rotation angle information of each joint of the mechanical arm in a joint space corresponding to an initial time and a termination time in an upper computer, and angular velocity and angular acceleration information of each joint of the mechanical arm corresponding to the time;
(4) differential state equation: based on the model method, the expression of the differential state equation of the mechanical arm is simplified by using a black box function, so that the application of the optimal control method to the optimal track planning problem of four-axis and above mechanical arms is realized, and the input and output of the differential state equation are u and x respectively 0 ,t,x,
y, the meanings represented by the variables and the corresponding relation with the mechanical arm system are as follows:
(1) u—input of differential state equation black box function, corresponding to output torque/current (τ/i) of each joint motor in the mechanical arm system;
②x 0 -initial state of the differential state equation black box function, corresponding to the rotation angle value (q 0 )；
(3) t-system run time with the initial state of the system as the initial time;
(4) x-state variable of the differential state equation black box function, corresponding to the rotation angle value and rotation angular velocity value of each joint in the mechanical arm system along with time change
⑤
-differentiation of the state variables of the differential state equation black box function, corresponding to the rotational angular velocity values, rotational angular acceleration values of the joints in the robotic arm system over time +.>
(6) y—the output of the differential state equation black box function, which can be defined as any variable in the system where it is set to correspond to the torque/current (τ/i) of each joint in the robotic arm system over time;
in summary, based on the model, the kinetic equation of the mechanical arm system is written as:
the differential state equation of the robotic arm system can be written as:
the solving technical route of the optimal control method is as follows:
(1) Initializing a mechanical arm system;
the method comprises the steps of inputting an initial gesture and a target gesture of the mechanical arm, and operating the mechanical arm to reach the initial gesture;
(2) Describing an optimal control problem;
the method comprises the steps of selecting performance indexes, establishing inequality constraints, giving boundary conditions and constructing differential state equation constraints of a mechanical arm system based on a model;
(3) Discretizing the optimal control problem by using a transcription method;
the method comprises the steps of converting an optimal control problem into a nonlinear programming problem by using a point matching method and a targeting method;
(4) And solving the nonlinear programming problem by using a nonlinear programming solver.
2. The optimal trajectory planning method for a model-based multi-degree of freedom mechanical arm according to claim 1, wherein the optimal trajectory planning method is characterized by: in the (4) of the optimal trajectory planning module, the differential state equation refers to the expression form of the system dynamics equation in the differential state space, and for the dynamics equation of the complex system, the expression form of the differential state space is difficult to analyze and deduce, so that a model-based method is used.
3. The optimal trajectory planning method for a model-based multi-degree of freedom mechanical arm according to claim 1, wherein the optimal trajectory planning method is characterized by: in the step (1) of the optimal trajectory planning module, four indexes of minimum time, minimum energy consumption, minimum impact and optimal mixing are respectively corresponding to improving the working efficiency of the mechanical arm, reducing the energy consumption of the mechanical arm and inhibiting vibration impact caused by abrupt change of the control quantity of the mechanical arm, and the optimal mixing index refers to optimizing the motion performance of the mechanical arm by comprehensively considering the effects of the indexes.
4. The optimal trajectory planning method for a model-based multi-degree of freedom mechanical arm according to claim 1, wherein the optimal trajectory planning method is characterized by: in the (1) of the optimal trajectory planning module, the mixed optimal four indexes are any two or more indexes of three indexes with minimum time, minimum energy consumption and minimum impact considered simultaneously.
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| CN108621158A (en) * | 2018-04-27 | 2018-10-09 | 上海师范大学 | A kind of time optimal trajectory planning control method and device about mechanical arm |
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| CN107490965A (en) * | 2017-08-21 | 2017-12-19 | 西北工业大学 | A kind of multiple constraint method for planning track of the free floating devices arm in space |
| CN108621158A (en) * | 2018-04-27 | 2018-10-09 | 上海师范大学 | A kind of time optimal trajectory planning control method and device about mechanical arm |
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