CN107351086B - Kalman estimation method for joint torque of SCARA robot - Google Patents

Kalman estimation method for joint torque of SCARA robot Download PDF

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CN107351086B
CN107351086B CN201710725626.2A CN201710725626A CN107351086B CN 107351086 B CN107351086 B CN 107351086B CN 201710725626 A CN201710725626 A CN 201710725626A CN 107351086 B CN107351086 B CN 107351086B
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CN107351086A (en
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张铁
梁骁翃
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South China University of Technology SCUT
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1633—Programme controls characterised by the control loop compliant, force, torque control, e.g. combined with position control

Abstract

The invention discloses a Kalman estimation method for joint torque of a SCARA robot, which comprises the following steps: step 1: obtaining a continuous nonlinear moment expression of each joint; step 2: obtaining a recursion linear discrete joint moment expression by utilizing a first-order Taylor expansion formula of a multivariate function; and step 3: taking the linear discretized expression of each joint moment as a process equation, adding a corresponding measurement equation, and combining and organizing the process equation and the measurement equation into a discrete time state space model of each joint moment; and 4, step 4: according to a basic Kalman filtering method, a formula required in a stepping cycle is determined by combining a discrete time state space model of joint torque, and the joint torque of the SCARA robot is estimated in real time by using read joint motor current signals and motor encoder signals. The invention improves the precision of joint moment estimation under the condition of not using a sensor, thereby providing reliable information for a robot control algorithm.

Description

Kalman estimation method for joint torque of SCARA robot
Technical Field
The invention relates to a method for estimating joint moment of a SCARA robot, in particular to a method for improving joint moment estimation precision based on a Kalman filtering method by taking a motor current signal with noise as basic information under the condition of not using a moment sensor.
Background
The realization of the high-speed and high-precision characteristics of the SCARA robot depends on an effective control algorithm, and the control algorithm has higher requirements on the precision of control information such as joint torque and the like. At present, the joint torque has two common acquisition modes, and is acquired by a torque sensor or converted by the current of a joint motor.
1) And acquiring joint torque through a torque sensor. The main problems with this approach are high cost and difficulty in installation and maintenance. Especially for some finished robots, if the design stage does not take into account the installation of sensors at the joints, the installation space may not be sufficient and the corresponding electrical interfaces may not be available.
2) And the joint torque is converted through the current of the joint motor. The method has the main problems that the current is analog quantity, noise exists, and the joint moment obtained by conversion has large fluctuation. The currently common method is to process the read data by hardware filtering or software filtering. However, the real-time performance of some filtering methods cannot be guaranteed, and therefore the implementation of a control algorithm is influenced.
Disclosure of Invention
The invention provides a Kalman estimation method aiming at SCARA robot joint moment, aiming at improving the precision of joint moment information required in a SCARA robot control algorithm under the condition of not using a moment sensor. The technical scheme is as follows:
a Kalman estimation method for SCARA robot joint torque comprises the following steps:
step 1: establishing a dynamic model of the SCARA robot according to a Newton Euler method to obtain a continuous non-linear moment expression of each joint;
step 2: converting a continuous nonlinear joint moment expression into a recursive linear discrete joint moment expression by using a first-order Taylor expansion formula of a multivariate function;
and step 3: taking the linear discretized expression of each joint moment as a process equation, adding a corresponding measurement equation, and combining and organizing the process equation and the measurement equation into a discrete time state space model of each joint moment;
and 4, step 4: according to a basic Kalman filtering method, a formula required in a stepping cycle is determined by combining a discrete time state space model of joint torque, and the joint torque of the SCARA robot is estimated in real time by using read joint motor current signals and motor encoder signals.
Further, in step 1, when a dynamic model of the SCARA robot is established, the third joint and the fourth joint of the SCARA robot are combined to form a combined connecting rod, and the combined joint is a mobile joint, so that the dynamic model of the SCARA robot comprises the first joint, the second joint and the third joint.
Further, in step 1, the continuous non-linear moment expression of each joint is as follows:
Figure BDA0001385947690000021
Figure BDA0001385947690000022
Figure BDA0001385947690000023
in the formula, a1，a2，a3，a4Is a coefficient item composed of parameters in a dynamically identifiable parameter set of the SCARA robot,the SCARA robot dynamics recognizable parameter set is obtained by early-stage parameter recognition work; f. of11，f12，f21，f22，f31，f32，P5，Jm3Is a part of the parameters in the identifiable parameter set, wherein f11，f21，f31Respectively representing the Coulomb friction coefficients of the first joint, the second joint and the third joint, f12，f22，f32Respectively representing the viscous friction coefficients of the first joint, the second joint and the third joint, P5As a combined parameter, Jm3The moment of inertia of the third joint motor; g is the acceleration of gravity, here only the magnitude, is-9.81 m.s-2；
Figure BDA0001385947690000024
Angular velocity and angular acceleration of the first joint, respectively;
Figure BDA0001385947690000025
angular velocity and angular acceleration of the second joint, respectively;
Figure BDA0001385947690000026
angular velocity and angular acceleration of the third joint, respectively.
Further, the step 2 specifically includes:
step 21: and respectively performing multivariate function first-order Taylor expansion on the continuous nonlinear moment expressions of all joints at the time k to obtain a linearized joint moment recursion formula:
Figure BDA0001385947690000031
Figure BDA0001385947690000032
Figure BDA0001385947690000033
in the formula, τ1,k，τ2,k，τ3,kRespectively representing joint force estimation moment values of a first joint, a second joint and a third joint at the moment k; tau is1,k+1，τ2,k+1，τ3,k+1Respectively representing joint moment estimated values of a first joint, a second joint and a third joint at the moment k + 1;
Figure BDA0001385947690000034
the angular velocity variation, angular acceleration variation, Δ q, of the first joint2，
Figure BDA0001385947690000035
The angle variation, the angular velocity variation and the angular acceleration variation of the second joint are respectively;
Figure BDA0001385947690000036
the angular velocity variation and the angular acceleration variation of the third joint are respectively;
Figure BDA0001385947690000037
the partial derivative of the first joint angular speed is obtained by representing a first joint moment expression, and the rest is similar;
Figure BDA0001385947690000038
the second joint moment expression is expressed to calculate the partial derivative of the first joint angular speed, and the rest is similar;
Figure BDA0001385947690000039
the expression of the third joint moment is used for solving the partial derivative of the third joint angular speed, and the rest is similar;
step 22: further rewriting the above linear expression can be obtained:
τ1,k+1＝τ1,k+Bu1,ku1,k+w1,k
τ2,k+1＝τ2,k+Bu2,ku2,k+w2,k
τ3,k+1＝τ3,k+Bu3,ku3,k+w3,k
in the formula (I), the compound is shown in the specification,
Figure BDA00013859476900000310
Figure BDA00013859476900000311
wherein
Figure BDA00013859476900000312
Figure BDA00013859476900000313
Figure BDA00013859476900000314
Wherein
Figure BDA00013859476900000315
Figure BDA0001385947690000041
Figure BDA0001385947690000042
Wherein
Figure BDA0001385947690000043
w1,k，w2,k，w3,kIs process noise.
Further, in step 3, the combination and arrangement of the process equation and the measurement equation into the discrete time state space model of each joint moment specifically includes:
state space model corresponding to the first joint:
τ1,k+1＝τ1,k+Bu1,ku1,k+w1,k
y1,k＝τ1,k+v1,k
state space model corresponding to the second joint:
τ2,k+1＝τ2,k+Bu2,ku2,k+w2,k
y2,k＝τ2,k+v2,k
the state space model corresponding to the third joint:
τ3,k+1＝τ3,k+Bu3,ku3,k+w3,k
y3,k＝τ3,k+v3,k
in the formula, v1,k，v2,k，v3,kTo measure noise.
Further, in step 4, the determining a formula required to be used in a step cycle according to the basic kalman filtering method by combining the discrete time state space model of the joint moment specifically includes:
step 41: let the discrete time state space model of the linear system be:
xk+1＝Fkxk+Bukuk+Bwkwk
yk＝Ckxk+vk
in the formula, xkIs a state vector; y iskIs a measurement vector; u. ofkIs an input vector; w is akIs process noise; v. ofkTo measure noise; fk，Buk，Bwk，CkIs a corresponding coefficient matrix; k in the subscript denotes the k time;
step 42: for a state space model in the form shown in the above equation, a step-loop kalman filter estimation can be performed according to the following equation:
Figure BDA0001385947690000051
Figure BDA0001385947690000052
Figure BDA0001385947690000053
Figure BDA0001385947690000054
Figure BDA0001385947690000055
in the formula, WkAnd VkCovariance matrices of process noise and metrology noise respectively,
Figure BDA0001385947690000056
and PtRespectively, the pre-test mean and the equation at time t,
Figure BDA0001385947690000057
and Pt' posterior means and equation at time t, respectively, KtIs the gain at the time t of the signal,
Figure BDA0001385947690000058
and Pt+1The pre-test mean and equation at time t +1, respectively.
Compared with the prior art, the invention has the advantages that: the method considers the characteristic of one side of the robot, comprehensively considers the motor current information and the dynamic factors of the robot when estimating the joint moment information of the SCARA robot without using a sensor, combines the motor current information with a dynamic model to estimate the joint moment, and deeply considers the actual application background compared with the simple application of hardware filtering or software filtering, thereby achieving the purpose of improving the moment estimation precision.
Drawings
Fig. 1 is a schematic diagram of a SCARA robot according to the present embodiment.
Shown in the figure: 1-a base; 2-a first articulated arm; 3-a second articulated arm; 4-a combined articulated arm of a third joint and a fourth joint.
Detailed Description
The invention will be further described with reference to the accompanying drawings and specific embodiments.
A Kalman estimation method for SCARA robot joint torque comprises the following steps:
step 1: establishing a dynamic model of the SCARA robot according to a Newton Euler method to obtain a continuous non-linear moment expression of each joint;
step 2: converting a continuous nonlinear joint moment expression into a recursive linear discrete joint moment expression by using a first-order Taylor expansion formula of a multivariate function;
and step 3: taking the linear discretized expression of each joint moment as a process equation, adding a corresponding measurement equation, and combining and organizing the process equation and the measurement equation into a discrete time state space model of each joint moment;
and 4, step 4: according to a basic Kalman filtering method, a formula required in a stepping cycle is determined by combining a discrete time state space model of joint torque, and the joint torque of the SCARA robot is estimated in real time by using read joint motor current signals and motor encoder signals.
Specifically, in step 1, when a dynamic model of the SCARA robot is created, as shown in fig. 1, the SCARA robot includes a base 1, a first joint arm 2, a second joint arm 3, and a combined joint arm 4 of a third joint and a fourth joint. The rotation motion of the fourth joint can cause a certain amount of same-direction coupling to the movement of the third joint, the third joint and the fourth joint of the SCARA robot are combined to form a combined connecting rod for reasonably simplifying a dynamic equation considering that the rotation motion of the fourth joint does not affect the main horizontal plane positioning motion of the SCARA robot, and the combined joint is a movable joint, so that the dynamic model of the SCARA robot comprises the first joint, the second joint and the third joint.
Specifically, the algorithm for calculating the joint moment using the newton euler method consists of two parts. The first part is to apply newton-euler equations to each link, iteratively calculating the velocity and acceleration of the links from the first link 1 to the nth link outward. The second part is to calculate the interaction force and moment between the connecting rods and the joint driving moment from the nth connecting rod to the 1 st connecting rod in an iterative way. Furthermore, the torque required by the rotor of the electric machine and the torque to overcome friction must be taken into account. The final total joint torque expression comprises three parts of driving torque, motor rotor torque and friction torque, so that in step 1, the continuous nonlinear torque expression of each joint is as follows:
Figure BDA0001385947690000061
Figure BDA0001385947690000062
Figure BDA0001385947690000063
in the formula, a1，a2，a3，a4The parameter set is a coefficient item consisting of parameters in a dynamic identifiable parameter set of the SCARA robot, and the dynamic identifiable parameter set of the SCARA robot is obtained by early-stage parameter identification work; f. of11，f12，f21，f22，f31，f32，P5，Jm3Is a part of the parameters in the identifiable parameter set, wherein f11，f21，f31Respectively representing the Coulomb friction coefficients of the first joint, the second joint and the third joint, f12，f22，f32Respectively representing the viscous friction coefficients of the first joint, the second joint and the third joint, P5As a combined parameter, Jm3The moment of inertia of the third joint motor; g is the acceleration of gravity, here only the magnitude, is-9.81 m.s-2；
Figure BDA0001385947690000071
Angular velocity and angular acceleration of the first joint, respectively;
Figure BDA0001385947690000072
angular velocity and angular acceleration of the second joint, respectively;
Figure BDA0001385947690000073
angular velocity and angular acceleration of the third joint, respectively.
Specifically, the step 2 specifically includes:
step 21: and respectively performing multivariate function first-order Taylor expansion on the continuous nonlinear moment expressions of all joints at the time k to obtain a linearized joint moment recursion formula:
Figure BDA0001385947690000074
Figure BDA0001385947690000075
Figure BDA0001385947690000076
in the formula, τ1,k，τ2,k，τ3,kRespectively representing joint force estimation moment values of a first joint, a second joint and a third joint at the moment k; tau is1,k+1，τ2,k+1，τ3,k+1Respectively representing joint moment estimated values of a first joint, a second joint and a third joint at the moment k + 1;
Figure BDA0001385947690000077
the angular velocity variation, angular acceleration variation, Δ q, of the first joint2，
Figure BDA0001385947690000078
The angle variation, the angular velocity variation and the angular acceleration variation of the second joint are respectively;
Figure BDA0001385947690000079
are respectively the third gateAngular velocity variation and angular acceleration variation of the nodes;
Figure BDA00013859476900000710
the partial derivative of the first joint angular speed is obtained by representing a first joint moment expression, and the rest is similar;
Figure BDA00013859476900000711
the second joint moment expression is expressed to calculate the partial derivative of the first joint angular speed, and the rest is similar;
Figure BDA00013859476900000712
the expression of the third joint moment is used for solving the partial derivative of the third joint angular speed, and the rest is similar;
step 22: further rewriting the above linear expression can be obtained:
τ1,k+1＝τ1,k+Bu1,ku1,k+w1,k
τ2,k+1＝τ2,k+Bu2,ku2,k+w2,k
τ3,k+1＝τ3,k+Bu3,ku3,k+w3,k
in the formula (I), the compound is shown in the specification,
Figure BDA0001385947690000081
wherein:
Figure BDA0001385947690000084
Figure BDA0001385947690000085
wherein
Figure BDA0001385947690000086
Figure BDA0001385947690000087
Figure BDA0001385947690000088
Wherein
Figure BDA0001385947690000089
w1,k，w2,k，w3,kIs process noise.
Specifically, in step 3, the combination and arrangement of the process equation and the measurement equation into the discrete time state space model of each joint moment specifically includes:
state space model corresponding to the first joint:
τ1,k+1＝τ1,k+Bu1,ku1,k+w1,k
y1,k＝τ1,k+v1,k
state space model corresponding to the second joint:
τ2,k+1＝τ2,k+Bu2,ku2,k+w2,k
y2,k＝τ2,k+v2,k
the state space model corresponding to the third joint:
τ3,k+1＝τ3,k+Bu3,ku3,k+w3,k
y3,k＝τ3,k+v3,k
in the formula, v1,k，v2,k，v3,kTo measure noise.
Specifically, in step 4, the determining a formula required to be used in a step cycle according to the basic kalman filtering method by combining the discrete time state space model of the joint moment specifically includes:
step 41: let the discrete time state space model of the linear system be:
xk+1＝Fkxk+Bukuk+Bwkwk
yk＝Ckxk+vk
in the formula, xkIs a state vector; y iskIs a measurement vector; u. ofkIs an input vector; w is akIs process noise; v. ofkTo measure noise; fk，Buk，Bwk，CkIs a corresponding coefficient matrix; k in the subscript denotes the k time;
step 42: for a state space model in the form shown in the above equation, a step-loop kalman filter estimation can be performed according to the following equation:
Figure BDA0001385947690000091
Figure BDA0001385947690000092
Figure BDA0001385947690000093
Figure BDA0001385947690000094
Figure BDA0001385947690000095
in the formula, WkAnd VkCovariance matrices of process noise and metrology noise respectively,
Figure BDA0001385947690000096
and PtRespectively, the pre-test mean and the equation at time t,
Figure BDA0001385947690000097
and Pt' posterior means and equation at time t, respectively, KtIs the gain at the time t of the signal,
Figure BDA0001385947690000098
and Pt+1The pre-test mean sum at time t +1And (4) an equation.
In summary, the estimation method provided by the invention firstly establishes a dynamic model of the SCARA robot according to a Newton Euler method to obtain a group of continuous nonlinear expressions of joint moments; then, performing discrete linearization on the joint moment expression by using first-order Taylor expansion of a multivariate function; then taking the discrete linear expression of the joint moment as a process equation, adding a corresponding measurement equation, combining and sorting two equations corresponding to each joint to obtain a discrete time state space model of the joint moment; and finally, by combining a state space model and a Kalman filtering method, the joint torque can be estimated in real time by using the motor current signal and the motor encoder signal read in each control period, and the real-time performance and the estimation precision can meet the requirements of the actual control of the robot.

Claims (6)

1. A Kalman estimation method for SCARA robot joint torque is characterized by comprising the following steps:
step 1: establishing a dynamic model of the SCARA robot according to a Newton Euler method to obtain a continuous non-linear moment expression of each joint;
step 2: converting a continuous nonlinear joint moment expression into a recursive linear discrete joint moment expression by using a first-order Taylor expansion formula of a multivariate function;
and step 3: taking the linear discretized expression of each joint moment as a process equation, adding a corresponding measurement equation, and combining and organizing the process equation and the measurement equation into a discrete time state space model of each joint moment;
and 4, step 4: according to a basic Kalman filtering method, a formula required in a stepping cycle is determined by combining a discrete time state space model of joint torque, and the joint torque of the SCARA robot is estimated in real time by using read joint motor current signals and motor encoder signals.
2. The method for kalman estimation of SCARA robot joint moment according to claim 1, characterized in that in step 1, when building a dynamical model of SCARA robot, the third joint and the fourth joint of SCARA robot are combined to form a combined link, and the combined joint is a mobile joint, so that the dynamical model of SCARA robot includes the first joint, the second joint and the third joint.
3. The method for kalman estimation of SCARA robot joint moments according to claim 2, characterized in that in step 1, the continuous non-linear moment expression of each joint is:
Figure FDA0002450962330000011
Figure FDA0002450962330000012
Figure FDA0002450962330000013
in the formula, a1，a2，a3，a4The parameter set is a coefficient item consisting of parameters in a dynamic identifiable parameter set of the SCARA robot, and the dynamic identifiable parameter set of the SCARA robot is obtained by early-stage parameter identification work; f. of11，f12，f21，f22，f31，f32，P5，Jm3Is a part of the parameters in the identifiable parameter set, wherein f11，f21，f31Respectively representing the Coulomb friction coefficients of the first joint, the second joint and the third joint, f12，f22，f32Respectively representing the viscous friction coefficients of the first joint, the second joint and the third joint, P5As a combined parameter, Jm3The moment of inertia of the third joint motor; g is the acceleration of gravity, here only the magnitude, is-9.81 m.s-2；
Figure FDA0002450962330000021
Angular velocity and angular acceleration of the first joint, respectively;
Figure FDA0002450962330000022
angular velocity and angular acceleration of the second joint, respectively;
Figure FDA0002450962330000023
angular velocity and angular acceleration of the third joint, respectively.
4. The SCARA robot joint moment Kalman estimation method according to claim 3, characterized in that the step 2 specifically comprises:
step 21: and respectively performing multivariate function first-order Taylor expansion on the continuous nonlinear moment expressions of all joints at the time k to obtain a linear off-line joint moment recurrence formula:
Figure FDA0002450962330000024
Figure FDA0002450962330000025
Figure FDA0002450962330000026
in the formula, τ1,k，τ2,k，τ3,kRespectively representing joint force estimation moment values of a first joint, a second joint and a third joint at the moment k; tau is1,k+1，τ2,k+1，τ3,k+1Respectively representing joint moment estimated values of a first joint, a second joint and a third joint at the moment k + 1;
Figure FDA0002450962330000027
the angular velocity variation, angular acceleration variation, Δ q, of the first joint2，
Figure FDA0002450962330000028
The angle variation, the angular velocity variation and the angular acceleration variation of the second joint are respectively;
Figure FDA0002450962330000029
the angular velocity variation and the angular acceleration variation of the third joint are respectively;
Figure FDA00024509623300000210
the partial derivative of the first joint angular speed is obtained by representing a first joint moment expression, and the rest is similar;
Figure FDA00024509623300000211
the second joint moment expression is expressed to calculate the partial derivative of the first joint angular speed, and the rest is similar;
Figure FDA00024509623300000212
the expression of the third joint moment is used for solving the partial derivative of the third joint angular speed, and the rest is similar;
step 22: further rewriting the above linear expression can be obtained:
τ1,k+1＝τ1,k+Bu1,ku1,k+w1,k
τ2,k+1＝τ2,k+Bu2,ku2,k+w2,k
τ3,k+1＝τ3,k+Bu3,ku3,k+w3,k
in the formula (I), the compound is shown in the specification,
Figure FDA0002450962330000031
Figure FDA0002450962330000032
wherein
Figure FDA0002450962330000033
Figure FDA0002450962330000034
Figure FDA0002450962330000035
Wherein
Figure FDA0002450962330000036
Figure FDA0002450962330000037
Figure FDA0002450962330000038
Wherein
Figure FDA0002450962330000039
w1,k，w2,k，w3,kIs process noise.
5. The method for kalman estimation of joint moments of SCARA robots according to claim 4, wherein in step 3, the combination of process equations and measurement equations into discrete time state space models of each joint moment specifically comprises:
state space model corresponding to the first joint:
τ1,k+1＝τ1,k+Bu1,ku1,k+w1,k
y1,k＝τ1,k+v1,k
state space model corresponding to the second joint:
τ2,k+1＝τ2,k+Bu2,ku2,k+w2,k
y2,k＝τ2,k+v2,k
the state space model corresponding to the third joint:
τ3,k+1＝τ3,k+Bu3,ku3,k+w3,k
y3,k＝τ3,k+v3,k
in the formula, v1,k，v2,k，v3,kTo measure noise.
6. The method for kalman estimation of joint moments of SCARA robots according to claim 5, wherein in step 4, the determining a formula needed in a step cycle according to the basic kalman filtering method in combination with the discrete time state space model of joint moments specifically comprises:
step 41: let the discrete time state space model of the linear system be:
xk+1＝Fkxk+Bukuk+Bwkwk
yk＝Ckxk+vk
in the formula, xkIs a state vector; y iskIs a measurement vector; u. ofkIs an input vector; w is akIs process noise; v. ofkTo measure noise; fk，Buk，Bwk，CkIs a corresponding coefficient matrix; k in the subscript denotes the k time;
step 42: for a discrete-time state-space model in the form shown in the above equation, a step-loop kalman filter estimation can be performed according to the following equation:
Figure FDA0002450962330000041
Figure FDA0002450962330000042
Figure FDA0002450962330000043
Figure FDA0002450962330000044
Figure FDA0002450962330000045
in the formula, WkAnd VkCovariance matrices of process noise and metrology noise respectively,
Figure FDA0002450962330000046
and PtRespectively, the pre-test mean and the equation at time t,
Figure FDA0002450962330000047
and Pt' posterior means and equation at time t, respectively, KtIs the gain at the time t of the signal,
Figure FDA0002450962330000048
and Pt+1The pre-test mean and equation at time t +1, respectively.
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