CN112989605B - Self-adaptive interactive impedance learning method for robot - Google Patents
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Abstract
The invention relates to a self-adaptive interaction impedance learning method for a robot, and belongs to the field of robot interaction force control; comprises a robot pin and a mechanical hole; the mechanical hole is a cylindrical through hole structure which is axially and horizontally arranged; the size of the mechanical hole corresponds to the size of the robot pin; the robot pin is arranged in the mechanical hole, and the outer wall of the robot pin is in contact with the inner wall of the mechanical hole; the robot pin moves along the axial direction of the mechanical hole; the robot pin is inserted into the mechanical hole at a constant speed, namely, the robot learns the impedance parameters of the environment, and the expected interaction dynamics characteristic is realized between the robot pin and the mechanical hole by adjusting the output force of the robot pin; the invention accurately reflects the characteristic that the impedance parameter changes along with the spatial position.
Description
Technical Field
The invention belongs to the field of robot interaction force control, and relates to a self-adaptive interaction impedance learning method for a robot.
Background
Impedance control is an effective method for controlling interaction force of a robot, and expected impedance parameters are often required to be given, so that an impedance controller is designed to control the robot to achieve expected interaction force. However, due to the unknowns and uncertainties of the external environment, it is often necessary to introduce iterative learning strategies to achieve adaptation to nonlinear factors such as damping and feedforward forces in the environment. Most current studies assume that the impedance parameters exhibit a periodic variation over time, based on which an impedance learning controller is designed. However, in typical interaction tasks such as precise assembly and plug-in connectors, physical parameters such as damping, surface roughness, elastic modulus and the like of a robot end tool and environment are related to relative positions of the robot end tool and the environment, but are not characteristics of periodic changes in a presentation domain, if the impedance learning mechanism based on time periodicity is still adopted, the characteristics of the impedance parameters changing along with the spatial position cannot be accurately reflected due to the movement speed change caused by the interaction force change.
Disclosure of Invention
The invention solves the technical problems that: the method overcomes the defects of the prior art, provides a self-adaptive interactive impedance learning algorithm of the robot, and accurately reflects the characteristic that impedance parameters change along with the spatial position.
The solution of the invention is as follows:
a self-adaptive interaction impedance learning method of a robot comprises the following steps:
step one, a self-adaptive interactive impedance system of a robot is established, wherein the self-adaptive interactive impedance system comprises a robot pin and a mechanical hole; the mechanical hole is a cylindrical through hole structure which is axially and horizontally arranged; the size of the mechanical hole corresponds to the size of the robot pin; the robot pin is arranged in the mechanical hole, and the outer wall of the robot pin is in contact with the inner wall of the mechanical hole; the robot pin moves along the axial direction of the mechanical hole.
A self-adaptive interaction impedance learning method of a robot comprises the following steps:
measuring parameters of the interactive impedance learning model, including the mass m of the robot pin and the driving force f of the robot pin u Environmental movement resistance f of robot pin e The moving distance x of the robot pin along the axial direction to the hole bottom of the mechanical hole;
step three, establishing an assembly mechanics formula of the robot pin and the mechanics hole:
in the formula ,the method is used for secondarily deriving the moving distance x of the robot pin along the axial direction to the bottom of the mechanical hole;
step four, setting the driving force of the robot pin for tracking the reference track as f r Setting the adaptive driving force for compensating the rest interaction force between the environment and the robot as f a The method comprises the steps of carrying out a first treatment on the surface of the Establishing robot pin driving force f u Is a calculation equation of (1);
fifthly, setting the hole depth of the mechanical hole as L; repeatedly extending the robot pin into the bottom of the mechanical hole for N times; calculating the accumulated moving distance l of the robot pin;
step six, setting the speed tracking error of the robot pin extending into the mechanical hole each time as epsilon i The method comprises the steps of carrying out a first treatment on the surface of the Wherein i is the number of times the robot pin extends into the mechanical hole; i is more than or equal to 1 and less than or equal to N; then the driving force f is adapted a The method comprises the following steps:
in the formula ,ka Is an adaptive force damping coefficient, i.e. k a ＝k d ；k d The damping coefficient is unknown and depends on the materials of the pin and the hole, the surface roughness, the lubrication condition and other factors;
the method is one-time derivation of the moving distance x of the robot pin along the axial direction to the bottom of the mechanical hole;
step seven, setting the position tracking error as e, wherein e=x-x r, wherein ,xr Is a reference track; the speed tracking error is wherein ,/>Is the reference speed; the driving force f of the robot pin to track the reference trajectory r The method comprises the following steps:
wherein alpha is a proportionality coefficient and is a positive value;
beta is a differential feedback gain coefficient and is a positive value;
the environmental movement resistance f of the robot pin e The method comprises the following steps:
in the formula ,kd The damping coefficient is unknown and depends on the materials of the pin and the hole, the surface roughness, the lubrication condition and other factors; the updated assembly mechanics formula is:
step eight, performing reduced-order processing on the formula (1) to obtain:
setting differential operators I.e. < ->
The equivalent kinetic model in the spatial domain is obtained from the second term of equation (2) as:
setting a conversion parameter z, z=y, and substituting the conversion parameter z into the formula (3):
where a (x) is an unknown coefficient vector, is periodic with x and is bounded,
a T (x) Transpose of the unknown coefficient vector a (x);
ζ 0 (z) is a known vector and is continuously derivable from x, z,
b (x) is an unknown bounded function with x being a periodic function, the function value being a positive number,
ρ (z) is an adaptive driving force coefficient related to z,
due toAccording to formula (3), deduce +.>Is represented by the expression:
wherein v (z, z r ) The method is a custom state conversion function;
step nine, deriving the self-adaptive damping coefficient k through a Lyapunov-Krasovskii function a The relationship with the axial movement distance x is as follows:
wherein, gamma is a feedback gain constant;
η (x) is the feedback gain, η=ce; c is a 1 xn constant vector, c= [ c ] 1 ,...,c n-1 ,1]The method comprises the steps of carrying out a first treatment on the surface of the e is a generalized velocity tracking error, which is an n×1-dimensional constant vector;
mu (x) is set as the extension parameter vector,an estimated value of the extended parameter vector;
setting zeta 0 (z) is a known vector that satisfies the local Lipschitz condition for x, and is continuously derivable for x, z;
zeta (x, e) is zeta 0 An expansion vector of (z); ζ (x, e) = [ (ζ) 0 ) T ,c 1 e-v(z,z r )] T； wherein ,c1 ＝[0,c 1 ,...,c n-1 ]To satisfy cp+c=c 1 P is the system matrix, and the system matrix is the system matrix,
set c= [ c ] 1 ,...,c n-1 ,1]To ensure that P is asymptotically stable;
step ten, in order to obtain the expected speed and stable interaction between the robot pin and the mechanical hole, a generalized speed tracking error e is defined as follows:
subtracting the formula (4) from the formula (5) to obtain an error dynamics equation of the closed-loop system:
where q= [0, ], 0,1] T ；
Then the control law equation:
wherein e is a generalized velocity tracking error, e= [ epsilon ] 1 ,...,ε n ] T ；
A p+2-dimensional diagonal matrix;
∈∈R p+2 the difference value between the expansion parameter vector and the estimation value of the expansion parameter vector;
then:
in the formula ,
introducing differential operatorsDifferentiating the formula (7):
introducing differential operatorsDifferentiating η=ce:
further:
where μ (x) is the extension parameter vector,
a (x) and b (x) are both periodic functions of x, and assuming that the period is L, then:
μ(l)＝μ(l-L)
when L is more than or equal to L,thus:
namely, when L is larger than or equal to L:
when L is more than or equal to 0 and less than or equal to L, then:
further:
at this time:
and η=ce, it can be seen that:
in the formula ,is about differential operator->Is a stable polynomial of (2);
from equation (9), η is bounded and converges with respect to L, so e is bounded and converges with respect to L, then the error dynamics equation of the closed loop systemThe solution exists, the error e is converged to 0 about L, namely, the robot can insert the robot pin into the mechanical hole at a constant speed, namely, the robot learns the impedance parameter of the environment, and the expected interaction dynamics characteristic is realized between the robot pin and the mechanical hole by adjusting the output force of the robot.
In the above-mentioned self-adaptive interactive impedance learning method for robot, in the third step, the driving force f u The calculation equation of (2) is:
f u ＝f r +f a 。
in the above-mentioned self-adaptive interaction impedance learning method for a robot, in the fourth step, the calculation method for the accumulated moving distance l is as follows:
l＝NL+x
wherein x is the moving distance x of the robot pin along the axial direction to the bottom of the mechanical hole for the last time after the robot pin is learned by NL.
In the above-mentioned robot adaptive interaction impedance learning method, in the eighth step, the estimated value of the parameter vector is expandedThe periodic update rule of (2) is:
wherein η (x) is the feedback gain;
Γ (L, L) is a learning gain matrix, and Γ (L, L) >0, Γ (L, L) has the value of:
wherein ,is a diagonal array;
is a diagonal array;
for l.epsilon.0, L), lambda i (x) Is a strictly increasing function and satisfies lambda i (0)＝0,λ i (L)＝ξ i ，ξ i >0 is a constant value.
Compared with the prior art, the invention has the beneficial effects that:
(1) The invention adopts the self-adaptive driving force f in the step five a The adjustment capability of the unknown environment is improved, and the fact that the pin is inserted into the hole according to expected dynamic characteristics of the robot in the unknown environment is realized;
(2) The invention adopts the step seven, namely, the differential operatorThe dynamic equation of the assembly process can be converted from a time domain to a space domain, and further the characteristic that the impedance parameter in space presents periodicity is adopted, and the impedance learning controller is designed, so that the method can reflect the interactive physical process more accurately;
(3) The book is provided withThe invention adopts the step eight, and the self-adaptive damping coefficient k a Through repeated iterative learning, the interaction force control capability of the robot to the uncertainty environment is improved, and the accurate learning of damping parameters of the unknown environment is realized.
Drawings
Fig. 1 is a schematic diagram of an interactive impedance learning device according to the present invention.
Detailed Description
The invention is further illustrated below with reference to examples.
The invention provides a self-adaptive interactive impedance learning method for a robot, which is used for realizing that a robot pin is inserted into a mechanical hole at a constant speed, namely, the robot learns the impedance parameters of the environment, and the expected interactive dynamics characteristic is realized between the robot pin and the mechanical hole by adjusting the output force of the robot pin; the invention accurately reflects the characteristic that the impedance parameter changes along with the spatial position.
The self-adaptive interaction impedance learning method of the robot comprises the following steps:
step one, a robot self-adaptive interaction impedance system is established, wherein the robot self-adaptive interaction impedance system comprises a robot pin 1 and a mechanical hole 2; as shown in fig. 1, the mechanical hole 2 is a cylindrical through hole structure horizontally arranged in the axial direction; the size of the mechanical hole 2 corresponds to the size of the robot pin 1; the robot pin 1 is arranged in the mechanical hole 2, and the outer wall of the robot pin 1 is in contact with the inner wall of the mechanical hole 2; the robot pin 1 moves along the axial direction of the mechanical hole 2.
Measuring parameters of the interactive impedance learning model, including the mass m of the robot pin 1 and the driving force f of the robot pin 1 u Environmental movement resistance f of robot pin 1 e The movement distance x of the robot pin 1 along the axial direction to the hole bottom of the mechanical hole 2.
Step three, establishing an assembly mechanics formula of the robot pin 1 and the mechanics hole 2:
in the formula ,the method is used for secondarily deriving the moving distance x of the robot pin 1 along the axial direction to the bottom of the mechanical hole 2.
Step four, setting the driving force of the robot pin 1 for tracking the reference track as f r Setting the adaptive driving force for compensating the rest interaction force between the environment and the robot as f a The method comprises the steps of carrying out a first treatment on the surface of the Establishing the robot pin 1 driving force f u Is calculated according to the equation: f (f) u ＝f r +f a 。
Step five, setting the hole depth of the mechanical hole 2 as L; repeatedly extending the robot pin 1 into the bottom of the mechanical hole 2 for N times; calculating the accumulated moving distance l of the robot pin 1; the calculation method of the accumulated moving distance l comprises the following steps:
l＝NL+x
in the formula, x is the moving distance x of the last time the robot pin 1 moves towards the bottom of the mechanical hole 2 along the axial direction after the robot pin 1 is learned by NL.
Step six, setting the tracking error of the speed of the robot pin 1 extending into the mechanical hole 2 each time as epsilon i The method comprises the steps of carrying out a first treatment on the surface of the Wherein i is the number of times that the robot pin 1 extends into the mechanical hole 2; i is more than or equal to 1 and less than or equal to N; then the driving force f is adapted a The method comprises the following steps:
in the formula ,ka Is an adaptive force damping coefficient, i.e. k a ＝k d ；k d The damping coefficient is unknown and depends on the materials of the pin and the hole, the surface roughness, the lubrication condition and other factors;
the method aims at once deriving the moving distance x of the robot pin 1 along the axial direction to the bottom of the mechanical hole 2.
Step seven, setting the position tracking error as e, wherein e=x-x r, wherein ,xr Is a reference track; the speed tracking error is wherein ,/>Is the reference speed; the driving force f of the robot pin 1 tracking the reference trajectory r The method comprises the following steps:
wherein alpha is a proportionality coefficient and is a positive value;
beta is a differential feedback gain coefficient and is a positive value;
the environmental movement resistance f of the robot pin 1 e The method comprises the following steps:
in the formula ,kd The damping coefficient is unknown and depends on the materials of the pin and the hole, the surface roughness, the lubrication condition and other factors; the updated assembly mechanics formula is:
step eight, performing reduced-order processing on the formula (1) to obtain:
setting differential operators I.e. < ->
The equivalent kinetic model in the spatial domain is obtained from the second term of equation (2) as:
setting a conversion parameter z, z=y, and substituting the conversion parameter z into the formula (3):
where a (x) is an unknown coefficient vector, is periodic with x and is bounded,
a T (x) Transpose of the unknown coefficient vector a (x);
ζ 0 (z) is a known vector and is continuously derivable from x, z,
b (x) is an unknown bounded function with x being a periodic function, the function value being a positive number,
ρ (z) is an adaptive driving force coefficient related to z,
due toAccording to formula (3), deduce +.>Is represented by the expression:
wherein v (z, z r ) Is a custom state transition function.
Step nine, deriving the self-adaptive damping coefficient k through a Lyapunov-Krasovskii function a The relationship with the axial movement distance x is as follows:
wherein, gamma is a feedback gain constant;
η (x) is the feedback gain, η=ce; c is a 1 xn constant vector, c= [ c ] 1 ,...,c n-1 ,1]The method comprises the steps of carrying out a first treatment on the surface of the e is a generalized velocity tracking error, which is an n×1-dimensional constant vector;
mu (x) is set as the extension parameter vector,an estimated value of the extended parameter vector;
setting zeta 0 (z) is a known vector that satisfies the local Lipschitz condition for x, and is continuously derivable for x, z;
zeta (x, e) is zeta 0 An expansion vector of (z); ζ (x, e) = [ (ζ) 0 ) T ,c 1 e-v(z,z r )] T； wherein ,c1 ＝[0,c 1 ,...,c n-1 ]To satisfy cp+c=c 1 P is the system matrix, and the system matrix is the system matrix,
set c= [ c ] 1 ,...,c n-1 ,1]To ensure that P is asymptotically stable.
Step ten, in order to obtain the expected speed and stable interaction between the robot pin 1 and the mechanical hole 2, a generalized speed tracking error e is defined as follows:
subtracting the formula (4) from the formula (5) to obtain an error dynamics equation of the closed-loop system:
where q= [0, ], 0,1] T ；
Then the control law equation:
wherein e is a generalized velocity tracking error, e= [ epsilon ] 1 ,...,ε n ] T ；
A p+2-dimensional diagonal matrix;
∈∈R p+2 the difference value between the expansion parameter vector and the estimation value of the expansion parameter vector;
then:
in the formula ,
introducing differential operatorsDifferentiating the formula (7):
introducing differential operatorsDifferentiating η=ce:
further:
where μ (x) is the extension parameter vector,
a (x) and b (x) are both periodic functions of x, and assuming that the period is L, then:
μ(l)＝μ(l-L)
when L is more than or equal to L,thus:
namely, when L is larger than or equal to L:
when L is more than or equal to 0 and less than or equal to L, then:/>
further:
at this time:
and η=ce, it can be seen that:
in the formula ,is about differential operator->Is a stable polynomial of (2);
from equation (9), η is bounded and converges with respect to L, so e is bounded and converges with respect to L, then the error dynamics equation of the closed loop systemThe solution is provided and the error e converges to 0 with respect to L, i.e. the robot is able to insert the robot pin 1 into the mechanical hole 2 at a constant speed, i.e. the robot learns the impedance parameters of the environment, by adjusting its own output force, the desired interaction dynamics between the robot pin 1 and the mechanical hole 2 is achieved.
Although the present invention has been described in terms of the preferred embodiments, it is not intended to be limited to the embodiments, and any person skilled in the art can make any possible variations and modifications to the technical solution of the present invention by using the methods and technical matters disclosed above without departing from the spirit and scope of the present invention, so any simple modifications, equivalent variations and modifications to the embodiments described above according to the technical matters of the present invention are within the scope of the technical matters of the present invention.
Claims (4)
1. A self-adaptive interaction impedance learning method of a robot is characterized in that: the method comprises the following steps:
step one, a self-adaptive interactive impedance system of a robot is established, wherein the self-adaptive interactive impedance system comprises a robot pin (1) and a mechanical hole (2); the mechanical hole (2) is a cylindrical through hole structure which is axially and horizontally arranged; the size of the mechanical hole (2) corresponds to the size of the robot pin (1); the robot pin (1) is arranged in the mechanical hole (2), and the outer wall of the robot pin (1) is in contact with the inner wall of the mechanical hole (2); the robot pin (1) moves along the axial direction of the mechanical hole (2);
measuring parameters of the interactive impedance learning model, including the mass m of the robot pin (1) and the driving force f of the robot pin (1) u Environmental movement resistance f of robot pin (1) e The moving distance x of the robot pin (1) along the axial direction to the hole bottom of the mechanical hole (2);
step three, establishing an assembly mechanics formula of the robot pin (1) and the mechanics hole (2):
in the formula ,the method is used for solving the secondary derivation of the moving distance x of the robot pin (1) to the bottom of the mechanical hole (2) along the axial direction;
step four, setting the driving force of the robot pin (1) for tracking the reference track as f r Setting the adaptive driving force for compensating the rest interaction force between the environment and the robot as f a The method comprises the steps of carrying out a first treatment on the surface of the Establishing the driving force f of the robot pin (1) u Is a calculation equation of (1);
step five, setting the hole depth of the mechanical hole (2) as L; repeatedly extending the robot pin (1) into the bottom of the mechanical hole (2) for N times; calculating the accumulated moving distance l of the robot pin (1);
step six, setting the tracking error of the speed of the robot pin (1) extending into the mechanical hole (2) each time as epsilon i The method comprises the steps of carrying out a first treatment on the surface of the Wherein i is the number of times the robot pin (1) extends into the mechanical hole (2); i is more than or equal to 1 and less than or equal to N; then the driving force f is adapted a The method comprises the following steps:
in the formula ,ka Is an adaptive force damping coefficient, i.e. k a ＝k d ；k d Is unknown damping coefficient and depends on the materials of the pin and the hole, the surface roughness and the lubrication condition factors;
the method is used for solving one-time derivation of the moving distance x of the robot pin (1) to the bottom of the mechanical hole (2) along the axial direction;
step seven, setting the position tracking error as e, wherein e=x-x r, wherein ,xr Is a reference track; the speed tracking error is wherein ,/>Is the reference speed; the driving force f of the robot pin (1) for tracking the reference trajectory r The method comprises the following steps:
wherein alpha is a proportionality coefficient and is a positive value;
beta is a differential feedback gain coefficient and is a positive value;
the environmental movement resistance f of the robot pin (1) e The method comprises the following steps:
in the formula ,kd Is unknown damping coefficient and depends on the materials of the pin and the hole, the surface roughness and the lubrication condition factors; the updated assembly mechanics formula is:
step eight, performing reduced-order processing on the formula (1) to obtain:
the differential operator is set to be equal to,i.e. < ->
The equivalent kinetic model in the spatial domain is obtained from the second term of equation (2) as:
setting a conversion parameter z, z=y, and substituting the conversion parameter z into the formula (3):
▽z＝a T (x)ζ 0 (z)+b(x)ρ(z)f a (4)
where a (x) is an unknown coefficient vector, is periodic with x and is bounded,
a T (x) Transpose of the unknown coefficient vector a (x);
ζ 0 (z) is a known vector and is continuously derivable from x, z,
b (x) is an unknown bounded function with x being a periodic function, the function value being a positive number,
ρ (z) is an adaptive driving force coefficient related to z,
due toDeriving% r Is represented by the expression:
▽z r ＝v(z,z r ) (5)
wherein v (z, z r ) The method is a custom state conversion function;
step nine, deriving the self-adaptive damping coefficient k through a Lyapunov-Krasovskii function a The relationship with the axial movement distance x is as follows:
wherein, gamma is a feedback gain constant;
η (x) is the feedback gain, η=ce; c is a 1 xn constant vector, c= [ c ] 1 ,...,c n-1 ,1]The method comprises the steps of carrying out a first treatment on the surface of the e is the generalized velocity tracking error and,is an n x 1-dimensional constant vector;
mu (x) is set as the extension parameter vector,an estimated value of the extended parameter vector;
setting zeta 0 (z) is a known vector that satisfies the local Lipschitz condition for x, and is continuously derivable for x, z;
zeta (x, e) is zeta 0 An expansion vector of (z); wherein ,c1 ＝[0,c 1 ,...,c n-1 ]To satisfy cp+c=c 1 P is the system matrix, and the system matrix is the system matrix,
set c= [ c ] 1 ,...,c n-1 ,1]To ensure that P is asymptotically stable;
step ten, in order to obtain the expected speed and stable interaction between the robot pin (1) and the mechanical hole (2), defining a generalized speed tracking error e as:
subtracting the formula (4) from the formula (5) to obtain an error dynamics equation of the closed-loop system:
▽e＝Pe+q(a T (x)ζ 0 +η(x)-v(z,z r )+b(x)ρ(z)f a )
where q= [0, ], 0,1] T ；
Then the control law equation:
wherein e is a generalized velocity tracking error, e= [ epsilon ] 1 ,...,ε n ] T ；
A p+2-dimensional diagonal matrix;
ε∈R p+2 the difference value between the expansion parameter vector and the estimation value of the expansion parameter vector;
then:
in the formula ,
introducing differential operatorsDifferentiating the formula (7):
introducing differential operatorsDifferentiating η=ce:
further:
where μ (x) is the extension parameter vector,
a (x) and b (x) are both periodic functions of x, and assuming that the period is L, then:
μ(l)＝μ(l-L)
when L is more than or equal to L,thus:
namely, when L is larger than or equal to L:
when L is more than or equal to 0 and less than or equal to L, then:
further:
at this time:
and η=ce, it can be seen that:
η＝ce
＝c 1 ε 1 +c 2 ε 2 +…+c n-1 ε n-1 +ε n
＝ε 1 (c 1 +c 2 ▽+…+c n-1 ▽ n-2 +▽ n-1 )
in the formula ,c1 +c 2 ▽+…+c n-1 ▽ n-2 +▽ n-1 A stabilizing polynomial with respect to the differential operator;
from equation (9), η is bounded and converges with respect to L, so e is bounded and converges with respect to L, and the error dynamics equation of the closed-loop system ∈=pe+q (a) T (x)ζ 0 +η(x)-v(z,z r )+b(x)ρ(z)f a ) The solution is provided, the error e is converged to 0 about L, namely, the robot can insert the robot pin (1) into the mechanical hole (2) at a constant speed, namely, the robot learns the impedance parameter of the environment, and the expected interaction dynamics characteristic is realized between the robot pin (1) and the mechanical hole (2) by adjusting the output force of the robot.
2. The method for learning self-adaptive interaction impedance of a robot according to claim 1, wherein: in the fourth step, the driving force f u The calculation equation of (2) is:
f u ＝f r +f a 。
3. the method for learning self-adaptive interaction impedance of a robot according to claim 2, wherein: in the fifth step, the method for calculating the accumulated moving distance l includes:
l＝NL+x
in the formula, x is the moving distance x of the last time the robot pin (1) moves towards the bottom of the mechanical hole (2) along the axial direction after the robot pin (1) is learned by NL.
4. A method for learning self-adaptive interaction impedance of a robot according to claim 3, wherein: in step nine, the estimated value of the extended parameter vectorThe periodic update rule of (2) is:
wherein η (x) is the feedback gain;
Γ (L, L) is a learning gain matrix, and Γ (L, L) >0, Γ (L, L) has the value of:
wherein ,is a diagonal array;
is a diagonal array;
for l.epsilon.0, L), lambda i (x) Is a strictly increasing function and satisfies lambda i (0)＝0,λ i (L)＝ξ i ，ξ i And >0 is a constant value.
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| CN109382826A (en) * | 2017-08-10 | 2019-02-26 | 精工爱普生株式会社 | Control device, robot and robot system |
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| CN111216130A (en) * | 2020-01-10 | 2020-06-02 | 电子科技大学 | Uncertain robot self-adaptive control method based on variable impedance control |
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| CN109382826A (en) * | 2017-08-10 | 2019-02-26 | 精工爱普生株式会社 | Control device, robot and robot system |
| CN108062024A (en) * | 2017-12-15 | 2018-05-22 | 陕西理工大学 | A kind of mobile robot back-stepping sliding mode control method for considering resistance |
| CN111037571A (en) * | 2019-12-31 | 2020-04-21 | 广东工业大学 | Robot self-adaptive variable damping impedance control method |
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