[足式机器人]Part4 南科大高等机器人控制课 CH12 Robotic Motion Control

发布时间:2023年12月29日

本文仅供学习使用
本文参考:
B站:CLEAR_LAB
笔者带更新-运动学
课程主讲教师:
Prof. Wei Zhang
课程链接 :
https://www.wzhanglab.site/teaching/mee-5114-advanced-control-for-robotics/


机器人——运动能力、计算能力、感知决策能力 的机电系统

1. Basic Linear Control Design

1.1 Error Response

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Steady-state error : e s s = lim ? t → ∞ θ e ( t ) e_{\mathrm{ss}}=\underset{t\rightarrow \infty}{\lim}\theta _{\mathrm{e}}\left( t \right) ess?=tlim?θe?(t)

Precent overshoot : P.O.

Rise time / Peak time :

Settling time : T s T_{\mathrm{s}} Ts?

1.2 Standard Second-Order Systems

详细推导见 : (待补充)
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1.3 Second-Order Response Characteristics

详细推导见 : (待补充)
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1.4 State-Space Controller Design

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  • Eigenvalue assignment : Find control gain K K K such that e i g ( A ? B K ) = e i g d e s i r e d eig\left( A-BK \right) =eig_{\mathrm{desired}} eig(A?BK)=eigdesired?
  • Solvability : We can always find such K K K if ( A , B ) \left( A,B \right) (A,B) is controllable ( r a n k ( m c ) = n rank\left( m_{\mathrm{c}} \right) =n rank(mc?)=n)
  • How to choose desired eigs? —— refer to 2nd-order system
    specification (P.O. T s T_{\mathrm{s}} Ts? T p T_{\mathrm{p}} Tp?) ? a r t \overset{art}{\Rightarrow} ?art dominant poles + other poles ? \Rightarrow ? e i g d e s i r e d eig_{\mathrm{desired}} eigdesired? ? s c i e n c e \overset{science}{\Rightarrow} ?science K K K

2. Motion Control Problems

2.1 Robotic Motion Control Problem

Dynamic equation of fully-acuated robot (with external force) : { τ = M ( q ) q ¨ + c ( q , q ˙ ) q ˙ + g ( q ) + J T ( q ) F e x t y = h ( q ) \begin{cases} \tau =M\left( q \right) \ddot{q}+c\left( q,\dot{q} \right) \dot{q}+g\left( q \right) +J^{\mathrm{T}}\left( q \right) \mathcal{F} _{\mathrm{ext}}\\ y=h\left( q \right)\\ \end{cases} {τ=M(q)q¨?+c(q,q˙?)q˙?+g(q)+JT(q)Fext?y=h(q)?
q ∈ R n q\in \mathbb{R} ^n qRn : joint positions (generalized coordinate)
τ ∈ R n \tau \in \mathbb{R} ^n τRn : joint torque (generalized input)
y y y : output (variable to be controlled) —— can be any func of q q q , e.g. y = q , y = [ T ( q ) ] ∈ S E ( 3 ) y=q,y=\left[ T\left( q \right) \right] \in SE\left( 3 \right) y=q,y=[T(q)]SE(3)

  • Motion Control Problems : Let y y y track given reference y d y_{\mathrm{d}} yd?
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    often times q d q_{\mathrm{d}} qd? is given by planner represented by polynomials , so that q ˙ d , q ¨ d \dot{q}_{\mathrm{d}},\ddot{q}_{\mathrm{d}} q˙?d?,q¨?d? can be easily obtained

2.2 Variations in Robot Motion Control

  • Joint-space vs. Task-space control
    Joint-space : y ( t ) = q ( t ) y\left( t \right) =q\left( t \right) y(t)=q(t) , i.e. , want q ( t ) q\left( t \right) q(t) to track a given q d ( t ) q_{\mathrm{d}}\left( t \right) qd?(t) joint reference
    Task-space : y ( t ) = [ T ( q ( t ) ) ] ∈ S E ( 3 ) y\left( t \right) =\left[ T\left( q\left( t \right) \right) \right] \in SE\left( 3 \right) y(t)=[T(q(t))]SE(3) denotes end-effector pose/configuration, we want y ( t ) y\left( t \right) y(t) to track y d ( t ) y_{\mathrm{d}}\left( t \right) yd?(t)

  • Actuation models:
    Velocity source : u = q ˙ u=\dot{q} u=q˙? —— directly control velocity
    Acceleration sources : u = q ¨ u=\ddot{q} u=q¨? —— directly control acceleration
    Torque sources : u = τ u=\tau u=τ —— directly control torque
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    Acutation model make sense if for ant given u u u , the joint velocity q ˙ \dot{q} q˙? can immediatly reach u u u

Motion Control Problem
Design u u u to set y y y track desired reference y d y_{\mathrm{d}} yd?

  • Depending on our assumption on u / y u/y u/y
    output y y y —— 6大基本问题
    y ? q ∈ R n y\leftrightarrow q\in \mathbb{R} ^n y?qRn - joint variable : Joint space motion control (Velocity-resolved Joint-space control ; Acceleration-resolved Joint-space control ; Torque-resolved Joint-space control ; )
    y ? [ T ( q ) ] ∈ S E ( 3 ) y\leftrightarrow \left[ T\left( q \right) \right] \in SE\left( 3 \right) y?[T(q)]SE(3) or y = f ( q ) y=f\left( q \right) y=f(q) - task space variable - e.g. origin of end-effector frame : Task space motion control (Velocity-resolved Task-space ; Acceleration-resolved Task-space ; Torque-resolved Task-space ; )

Linear control / feedback lineariazation

3. Motion Control with Velocity/Acceleration as Input

3.1 Velocity-Resolved Control

Each joints’ velocity q ˙ i \dot{q}_{\mathrm{i}} q˙?i? can be directly controlled

Good approximation for hydraulic actuators

Common approxiamtion of the outer-loop control for the Inner / outer loop control setup
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3.2.1 Velocity-Resolved Joint Space Control

Joint-space ‘dynamics’ : single integrator q ˙ = u \dot{q}=u q˙?=u

Joint-space tracking becomes standard linear tracking control problem : u = q ˙ d + K 0 q ¨ ? q ~ ˙ + K 0 q ¨ = 0 u=\dot{q}_{\mathrm{d}}+K_0\ddot{q}\Rightarrow \dot{\tilde{q}}+K_0\ddot{q}=0 u=q˙?d?+K0?q¨??q~?˙?+K0?q¨?=0 , where q ~ = q d ? q \tilde{q}=q_{\mathrm{d}}-q q~?=qd??q is the joint position error. —— stable if e i g ( ? K 0 ) ∈ O L H P eig\left( -K_0 \right) \in OLHP eig(?K0?)OLHP

The error dynamic is stable if ? K 0 -K_0 ?K0? is Hurwitz

3.2.2 Velocity-Resolved Task Space Control

For task space control , y = [ T ( q ) ] y=\left[ T\left( q \right) \right] y=[T(q)] needs to track y d y_{\mathrm{d}} yd? , y y y can be ant function of q q q, in particular , it can represents position and/or the end-effector frame

Taking derivatives of y y y , and letting u = q ˙ u=\dot{q} u=q˙? , we have : y ˙ = J a ( q ) u \dot{y}=J_{\mathrm{a}}\left( q \right) u y˙?=Ja?(q)u
Note that q q q is function of y y y through inverse kinematics ( q = I K ( y ) q=IK\left( y \right) q=IK(y))
So the above dynamics can be written in terms of y y y and u u u only. The detailed form can be quite complex in general y ˙ = J a ( I K ( y ) ) u \dot{y}=J_{\mathrm{a}}\left( IK\left( y \right) \right) u y˙?=Ja?(IK(y))u

  1. Let v y v_{\mathrm{y}} vy? be virtual control y ˙ = v y \dot{y}=v_{\mathrm{y}} y˙?=vy? design v y v_{\mathrm{y}} vy? to track y d y_{\mathrm{d}} yd? (same as above)
  2. Find actual control u u u such that J a ( I K ( y ) ) u ≈ v y J_{\mathrm{a}}\left( IK\left( y \right) \right) u\approx v_{\mathrm{y}} Ja?(IK(y))uvy?

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We can design outer-loop controller as if we can directly control y ˙ \dot{y} y˙?
y ˙ = v y = y ˙ d + K ( y d ? y ) ? p l u g ?? i n ?? y ˙ = v y ?? y ~ ˙ = ? K y ~ \dot{y}=v_{\mathrm{y}}=\dot{y}_{\mathrm{d}}+K\left( y_{\mathrm{d}}-y \right) \overset{plug\,\,in\,\,\dot{y}=v_{\mathrm{y}}\,\,}{\Longrightarrow}\dot{\tilde{y}}=-K\tilde{y} y˙?=vy?=y˙?d?+K(yd??y)?pluginy˙?=vy??y~?˙?=?Ky~?
We can select K K K such that ? K -K ?K is Hurtwiz , object of inner loop : determine u = q ˙ u=\dot{q} u=q˙? such that y ˙ ≈ v y \dot{y}\approx v_{\mathrm{y}} y˙?vy?

System(2) is nonlinear system , a commeon way is to break it into inner-outer loop , where the outer loop directly control velocity of y y y, and the inner loop tries to find u u u to generate desired task space velocity

Outer loop : y ˙ = v y \dot{y}=v_{\mathrm{y}} y˙?=vy? , where control v y = y ˙ d + K 0 y ~ v_{\mathrm{y}}=\dot{y}_{\mathrm{d}}+K_0\tilde{y} vy?=y˙?d?+K0?y~? , resulting in task-space closed-loop error dynamics: y ~ ˙ + K 0 y ~ = 0 \dot{\tilde{y}}+K_0\tilde{y}=0 y~?˙?+K0?y~?=0

Above task space tracking relies on a fictitious control v y v_{\mathrm{y}} vy? , i.e. , it assumes y ˙ \dot{y} y˙? can be arbitrarily controlled by selecting appropriate u = q ˙ u=\dot{q} u=q˙? , which is true if J a J_{\mathrm{a}} Ja? is full-row rank

Inner loop : Given v y v_{\mathrm{y}} vy? from the outer loop, find the joint velocity control by solving
{ min ? u ∥ v y ? J a ( q ) u ∥ 2 + r e g u l a r i z a t i o n ?? t e r m s u b j . t o ?? : C o n s t r a i n t s ?? o n ?? u ?? , e . g . { q ˙ min ? ? u ? q ˙ max ? q min ? ? q + u Δ t ? q max ? \begin{cases} \min _{\mathrm{u}}\left\| v_{\mathrm{y}}-J_{\mathrm{a}}\left( q \right) u \right\| ^2+regularization\,\,term\\ subj.to\,\,: Constraints\,\,on\,\,u\,\,, e.g.\begin{cases} \dot{q}_{\min}\leqslant u\leqslant \dot{q}_{\max}\\ q_{\min}\leqslant q+u\varDelta t\leqslant q_{\max}\\ \end{cases}\\ \end{cases} ? ? ??minu?vy??Ja?(q)u2+regularizationtermsubj.to:Constraintsonu,e.g.{q˙?min??u?q˙?max?qmin??q+uΔt?qmax???
Inner-loop is essentially a differential IK controller
One can also use the pseudo-inverse control u = J a ? v y u={J_{\mathrm{a}}}^{\dagger}v_{\mathrm{y}} u=Ja??vy?

3.2 Acceleration-Resolved Control

3.2.1 Acceleration-Resolved Control in Joint Space

Joint acceleration cna be directly controlled , resulting in double-integrator dynamics q ¨ = u \ddot{q}=u q¨?=u . Given q d q_{\mathrm{d}} qd? reference , we want q → q d q\rightarrow q_{\mathrm{d}} qqd? (double integartor)

Joint-space tracking becomes standard linear tracking control problem for double-integrator system:
u = q ¨ d + K 1 q ~ ˙ + K 0 q ~ = 0 , q ~ ∈ R n u=\ddot{q}_{\mathrm{d}}+K_1\dot{\tilde{q}}+K_0\tilde{q}=0,\tilde{q}\in \mathbb{R} ^n u=q¨?d?+K1?q~?˙?+K0?q~?=0,q~?Rn
—— PD control , closed-loop system , where q ~ = q d ? q \tilde{q}=q_{\mathrm{d}}-q q~?=qd??q is the joint position error.

Stablility condition : Let x = [ q ~ q ~ ˙ ] ∈ R 2 n x=\left[ \begin{array}{c} \tilde{q}\\ \dot{\tilde{q}}\\ \end{array} \right] \in \mathbb{R} ^{2n} x=[q~?q~?˙??]R2n , [ 0 E ? K 0 ? K 1 ] [ q ~ q ~ ˙ ] , x ˙ = A x \left[ \begin{matrix} 0& E\\ -K_0& -K_1\\ \end{matrix} \right] \left[ \begin{array}{c} \tilde{q}\\ \dot{\tilde{q}}\\ \end{array} \right] ,\dot{x}=Ax [0?K0??E?K1??][q~?q~?˙??],x˙=Ax
closed-loop system is stable . if e i g ( A ) ∈ O L H P eig\left( A \right) \in OLHP eig(A)OLHP or A A A is Hurwitz

3.2.2 Acceleration-Resolved Control in Task Space

For task space control , y = [ T ( q ) ] ∈ S E ( 3 ) y=\left[ T\left( q \right) \right] \in SE\left( 3 \right) y=[T(q)]SE(3) needs to track y d y_{\mathrm{d}} yd?
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Note : For y = f ( q ) y=f\left( q \right) y=f(q) y ˙ = J a ( q ) q ˙ \dot{y}=J_{\mathrm{a}}\left( q \right) \dot{q} y˙?=Ja?(q)q˙? and y ¨ = J ˙ a ( q ) q ˙ + J a ( q ) q ¨ ? y ¨ = J ˙ a ( q ) q ˙ + J a ( q ) u ? \ddot{y}=\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}+J_{\mathrm{a}}\left( q \right) \ddot{q}\Rightarrow \ddot{y}=\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}+J_{\mathrm{a}}\left( q \right) u\Leftarrow y¨?=J˙a?(q)q˙?+Ja?(q)q¨??y¨?=J˙a?(q)q˙?+Ja?(q)u? nonlinear dynamics

Following the same inner-outer loop strategy deiscussed before . Introduce virtual control , a y a_{\mathrm{y}} ay? such that y ¨ = a y \ddot{y}=a_{\mathrm{y}} y¨?=ay? , we can design controller for a y a_{\mathrm{y}} ay? to let y → y d y\rightarrow y_{\mathrm{d}} yyd?

Outer-loop dynamics : y ¨ = a y \ddot{y}=a_{\mathrm{y}} y¨?=ay? , with a y a_{\mathrm{y}} ay? being the outer-loop control input a y = y ¨ d + K 1 y ~ ˙ + K 0 y ~ ? y ~ ¨ + K 1 y ~ ˙ + K 0 y ~ = 0 a_{\mathrm{y}}=\ddot{y}_{\mathrm{d}}+K_1\dot{\tilde{y}}+K_0\tilde{y}\Rightarrow \ddot{\tilde{y}}+K_1\dot{\tilde{y}}+K_0\tilde{y}=0 ay?=y¨?d?+K1?y~?˙?+K0?y~??y~?¨?+K1?y~?˙?+K0?y~?=0
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—— PD control , stable if [ 0 E ? K 0 ? K 1 ] \left[ \begin{matrix} 0& E\\ -K_0& -K_1\\ \end{matrix} \right] [0?K0??E?K1??] Hurwitz

Inner-loop : given a y a_{\mathrm{y}} ay? from outer loop , find the “best” joint acceleration:
{ min ? u ∥ a y ? J ˙ a ( q ) q ˙ ? J a ( q ) u ∥ 2 + r e g u l a r i z a t i o n ?? t e r m s u b j . t o ?? : C o n s t r a i n t s ?? o n ?? u ?? \begin{cases} \min _{\mathrm{u}}\left\| a_{\mathrm{y}}-\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}-J_{\mathrm{a}}\left( q \right) u \right\| ^2+regularization\,\,term\\ subj.to\,\,: Constraints\,\,on\,\,u\,\,\\ \end{cases} ? ? ??minu? ?ay??J˙a?(q)q˙??Ja?(q)u ?2+regularizationtermsubj.to:Constraintsonu?
—— u u u : optimization variable , J ˙ a ( q ) , q ˙ , q \dot{J}_{\mathrm{a}}\left( q \right) ,\dot{q},q J˙a?(q),q˙?,q - known
{ A c c ?? : q ¨ min ? ? u ? q ¨ max ? V e l ?? : q ˙ min ? ? q + u Δ t ? q ˙ max ? \begin{cases} Acc\,\,: \ddot{q}_{\min}\leqslant u\leqslant \ddot{q}_{\max}\\ Vel\,\,: \dot{q}_{\min}\leqslant q+u\varDelta t\leqslant \dot{q}_{\max}\\ \end{cases} {Acc:q¨?min??u?q¨?max?Vel:q˙?min??q+uΔt?q˙?max??

Mathematically , the above problem is the same as the Differential IK problem

At any given time , q ˙ , q \dot{q},q q˙?,q can be measured , and then y , y ˙ y,\dot{y} y,y˙? can be computed, which allows us to compute outer loop control a y a_{\mathrm{y}} ay? and inner loop control u u u

4. Motion Control with Torque as Input and Task Space Inverse Dynamics

4.1 Recall Properties of Robot Dynamics

For fully actuated robot :
τ = M ( q ) q ¨ + C ( q , q ˙ ) q ˙ + g ( q ) \tau =M\left( q \right) \ddot{q}+C\left( q,\dot{q} \right) \dot{q}+g\left( q \right) τ=M(q)q¨?+C(q,q˙?)q˙?+g(q)
M ( q ) = ∑ J i T [ I i ] 6 × 6 J i ∈ R n × n M\left( q \right) =\sum{{J_{\mathrm{i}}}^{\mathrm{T}}\left[ \mathcal{I} _{\mathrm{i}} \right] _{6\times 6}J_{\mathrm{i}}}\in \mathbb{R} ^{n\times n} M(q)=Ji?T[Ii?]6×6?Ji?Rn×n

There are many valid difinitions of C ( q , q ˙ ) C\left( q,\dot{q} \right) C(q,q˙?) , typical choice for C C C include:
C i j = ∑ k 1 2 ( ? M i j ? q k + ? M i k ? q j ? ? M j k ? q i ) C_{\mathrm{ij}}=\sum_k^{}{\frac{1}{2}\left( \frac{\partial M_{\mathrm{ij}}}{\partial q_{\mathrm{k}}}+\frac{\partial M_{\mathrm{ik}}}{\partial q_{\mathrm{j}}}-\frac{\partial M_{\mathrm{jk}}}{\partial q_{\mathrm{i}}} \right)} Cij?=k?21?(?qk??Mij??+?qj??Mik????qi??Mjk??)
For the above defined C C C , we have M ˙ ? 2 C \dot{M}-2C M˙?2C is skew symmetric
For all valid C C C, we have q ˙ T [ M ˙ ? 2 C ] q ˙ = 0 \dot{q}^{\mathrm{T}}\left[ \dot{M}-2C \right] \dot{q}=0 q˙?T[M˙?2C]q˙?=0
These properties play improtant role in designing motion controller

4.2 Computed Torque Control

For fully-actuated robot, we have M ( q ) ? 0 M\left( q \right) \succ 0 M(q)?0 and q ¨ \ddot{q} q¨? can be arbitrarily specified through torque control u = τ u=\tau u=τ
q ¨ = M ? 1 ( q ) [ u ? C ( q , q ˙ ) q ˙ ? g ( q ) ] \ddot{q}=M^{-1}\left( q \right) \left[ u-C\left( q,\dot{q} \right) \dot{q}-g\left( q \right) \right] q¨?=M?1(q)[u?C(q,q˙?)q˙??g(q)]

we know how to design controller if u = q ¨ u=\ddot{q} u=q¨?
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Thus , for fully-acuated robot, torque controlled case can be reduced to the acceleration-resolved case

Outer loop: q ¨ = a q \ddot{q}=a_{\mathrm{q}} q¨?=aq? with joint acceleration as control input
a q = q ¨ + K 1 y ~ ˙ + K 0 y ~ ? q ~ ¨ + K 1 q ~ ˙ + K 0 q ~ = 0 a_{\mathrm{q}}=\ddot{q}+K_1\dot{\tilde{y}}+K_0\tilde{y}\Rightarrow \ddot{\tilde{q}}+K_1\dot{\tilde{q}}+K_0\tilde{q}=0 aq?=q¨?+K1?y~?˙?+K0?y~??q~?¨?+K1?q~?˙?+K0?q~?=0

Inner loop : since M ( q ) M\left( q \right) M(q) is square and nonsingular , inner loop control u u u can be found analytically:
u = M ( q ) ( q ¨ d + K 1 q ~ ˙ + K 0 q ~ ) + C ( q , q ˙ ) q ˙ + g ( q ) u=M\left( q \right) \left( \ddot{q}_{\mathrm{d}}+K_1\dot{\tilde{q}}+K_0\tilde{q} \right) +C\left( q,\dot{q} \right) \dot{q}+g\left( q \right) u=M(q)(q¨?d?+K1?q~?˙?+K0?q~?)+C(q,q˙?)q˙?+g(q)
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The control law is a function of q , q ˙ q,\dot{q} q,q˙? and the reference q d q_{\mathrm{d}} qd?. It is called computed-torque control.

The control law also relies on system model M , C , g M,C,g M,C,g if these model information are not accurate, the control will not perform well.
y = f ( q ) , y ¨ = J ˙ a ( q ) q ˙ + J a ( q ) M ? 1 ( u ? C ? g ) y=f\left( q \right) ,\ddot{y}=\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}+J_{\mathrm{a}}\left( q \right) M^{-1}\left( u-C-g \right) y=f(q),y¨?=J˙a?(q)q˙?+Ja?(q)M?1(u?C?g)
Idea easily extends to task space : y ˙ = J a ( q ) q ˙ \dot{y}=J_{\mathrm{a}}\left( q \right) \dot{q} y˙?=Ja?(q)q˙? and y ¨ = J ˙ a ( q ) q ˙ + J a ( q ) q ¨ \ddot{y}=\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}+J_{\mathrm{a}}\left( q \right) \ddot{q} y¨?=J˙a?(q)q˙?+Ja?(q)q¨? —— τ = u = τ , q ¨ = M ? 1 [ u ? C ? g ] \tau =u=\tau ,\ddot{q}=M^{-1}\left[ u-C-g \right] τ=u=τ,q¨?=M?1[u?C?g]

Outer loop : y ¨ = a y \ddot{y}=a_{\mathrm{y}} y¨?=ay? and a y = y ¨ d + K 1 y ~ ˙ + K 0 y ~ a_{\mathrm{y}}=\ddot{y}_{\mathrm{d}}+K_1\dot{\tilde{y}}+K_0\tilde{y} ay?=y¨?d?+K1?y~?˙?+K0?y~?

Inner loop : sekect torque control u = τ u=\tau u=τ by
{ min ? u ∥ a y ? J ˙ a ( q ) q ˙ ? J a ( q ) M ? 1 ( u ? C q ˙ ? g ) ∥ 2 s u b j . t o ?? : C o n s t r a i n t s ?? \begin{cases} \min _{\mathrm{u}}\left\| a_{\mathrm{y}}-\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}-J_{\mathrm{a}}\left( q \right) M^{-1}\left( u-C\dot{q}-g \right) \right\| ^2\\ subj.to\,\,: Constraints\,\,\\ \end{cases} ? ? ??minu? ?ay??J˙a?(q)q˙??Ja?(q)M?1(u?Cq˙??g) ?2subj.to:Constraints?
If J a J_{\mathrm{a}} Ja?is invertible and we don’t impose additional torque constraints, analytical control law can be easily obtained —— u = ( J a ( q ) M ? 1 ) ? 1 ( a y ? J ˙ a ( q ) q ˙ . . . ) u=\left( J_{\mathrm{a}}\left( q \right) M^{-1} \right) ^{-1}\left( a_{\mathrm{y}}-\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}... \right) u=(Ja?(q)M?1)?1(ay??J˙a?(q)q˙?...)

4.3 Inverse Dynamics Control

The computed-torque controller above is also canned inverse dynamics control

Forward dynamics : given τ \tau τ to compute q ¨ \ddot{q} q¨? —— from torque to motion

Inverse dynamics : given desired acceleration a q a_{\mathrm{q}} aq?, we inverted it to find the required control by u = M a q + C q ˙ + g u=Ma_{\mathrm{q}}+C\dot{q}+g u=Maq?+Cq˙?+g

Task space case can be viewed as inverting the task space dynamics —— Given a y a_{\mathrm{y}} ay? ( y y y task space) , find τ \tau τ such that y ¨ = a y \ddot{y}=a_{\mathrm{y}} y¨?=ay?

With recent advances in optimization , it is often preferred to do ID with quedratic program
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For example, above equation can be viewed as task-space ID. We can incorporate torque contraints explicitly as follows:
{ min ? u ∥ a y ? J ˙ a ( q ) q ˙ ? J a M ? 1 ( u ? C q ˙ ? g ) ∥ 2 s u b j . t o ?? : u ? ? u ?? ? u + ?? \begin{cases} \min _{\mathrm{u}}\left\| a_{\mathrm{y}}-\dot{J}_{\mathrm{a}}\left( q \right) \dot{q}-J_{\mathrm{a}}M^{-1}\left( u-C\dot{q}-g \right) \right\| ^2\\ subj.to\,\,: u_-\leqslant u\,\,\leqslant u_+\,\,\\ \end{cases} ? ? ??minu? ?ay??J˙a?(q)q˙??Ja?M?1(u?Cq˙??g) ?2subj.to:u???u?u+??
optimization variable u ∈ R n u\in \mathbb{R} ^n uRn

This is equivalent to the following more popular form:
{ min ? u , q ¨ ∥ a y ? J ˙ a q ˙ ? J a q ¨ ∥ 2 s u b j . t o ?? : M q ¨ + C q ˙ + g = u u ? ? u ∈ R n ?? ? u + ?? \begin{cases} \underset{u,\ddot{q}}{\min}\left\| a_{\mathrm{y}}-\dot{J}_{\mathrm{a}}\dot{q}-J_{\mathrm{a}}\ddot{q} \right\| ^2\\ subj.to\,\,: \begin{array}{c} M\ddot{q}+C\dot{q}+g=u\\ u_-\leqslant u\in \mathbb{R} ^n\,\,\leqslant u_+\,\,\\ \end{array}\\ \end{cases} ? ? ??u,q¨?min? ?ay??J˙a?q˙??Ja?q¨? ?2subj.to:Mq¨?+Cq˙?+g=uu???uRn?u+???
optimization variable u , q ¨ ∈ R n u,\ddot{q}\in \mathbb{R} ^n u,q¨?Rn

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