网站怎么优化,陕西网站备案查询,百度代理查询,做健康类网站怎么备案本文仅供学习使用 本文参考#xff1a; B站#xff1a;CLEAR_LAB 笔者带更新-运动学 课程主讲教师#xff1a; Prof. Wei Zhang 课程链接 #xff1a; https://www.wzhanglab.site/teaching/mee-5114-advanced-control-for-robotics/ 南科大高等机器人控制课 Ch12 Robotic … 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/ 南科大高等机器人控制课 Ch12 Robotic Motion Control 1. Basic Linear Control Design1.1 Error Response1.2 Standard Second-Order Systems1.3 Second-Order Response Characteristics1.4 State-Space Controller Design 2. Motion Control Problems2.1 Robotic Motion Control Problem2.2 Variations in Robot Motion Control 3. Motion Control with Velocity/Acceleration as Input3.1 Velocity-Resolved Control3.2.1 Velocity-Resolved Joint Space Control3.2.2 Velocity-Resolved Task Space Control 3.2 Acceleration-Resolved Control3.2.1 Acceleration-Resolved Control in Joint Space3.2.2 Acceleration-Resolved Control in Task Space 4. Motion Control with Torque as Input and Task Space Inverse Dynamics4.1 Recall Properties of Robot Dynamics4.2 Computed Torque Control4.3 Inverse Dynamics Control 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 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) esst→∞limθe(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
1.3 Second-Order Response Characteristics
详细推导见 待补充
1.4 State-Space Controller Design 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)eigdesiredSolvability : 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}}\\ yh\left( q \right)\\ \end{cases} {τM(q)q¨c(q,q˙)q˙g(q)JT(q)Fextyh(q) q ∈ R n q\in \mathbb{R} ^n q∈Rn : joint positions (generalized coordinate) τ ∈ R n \tau \in \mathbb{R} ^n τ∈Rn : 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 ) yq,y\left[ T\left( q \right) \right] \in SE\left( 3 \right) yq,y[T(q)]∈SE(3)
Motion Control Problems : Let y y y track given reference y d y_{\mathrm{d}} yd 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} uq˙ —— directly control velocity Acceleration sources : u q ¨ u\ddot{q} uq¨ —— directly control acceleration Torque sources : u τ u\tau uτ —— directly control torque 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↔q∈Rn - 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 ) yf\left( q \right) yf(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
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 uq˙dK0q¨⇒q~˙K0q¨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} uq˙ , 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 ) qIK\left( y \right) qIK(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
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)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))u≈vy 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˙vyy˙dK(yd−y)⟹pluginy˙vyy~˙−Ky~ We can select K K K such that − K -K −K is Hurtwiz , object of inner loop : determine u q ˙ u\dot{q} uq˙ 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} vyy˙dK0y~ , resulting in task-space closed-loop error dynamics: y ~ ˙ K 0 y ~ 0 \dot{\tilde{y}}K_0\tilde{y}0 y~˙K0y~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} uq˙ , 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\| ^2regularization\,\,term\\ subj.to\,\,: Constraints\,\,on\,\,u\,\,, e.g.\begin{cases} \dot{q}_{\min}\leqslant u\leqslant \dot{q}_{\max}\\ q_{\min}\leqslant qu\varDelta t\leqslant q_{\max}\\ \end{cases}\\ \end{cases} ⎩ ⎨ ⎧minu∥vy−Ja(q)u∥2regularizationtermsubj.to:Constraintsonu,e.g.{q˙min⩽u⩽q˙maxqmin⩽quΔt⩽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}} uJa†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}} q→qd (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 uq¨dK1q~˙K0q~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−K0E−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 Note : For y f ( q ) yf\left( q \right) yf(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}} y→yd
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 ayy¨dK1y~˙K0y~⇒y~¨K1y~˙K0y~0 —— PD control , stable if [ 0 E − K 0 − K 1 ] \left[ \begin{matrix} 0 E\\ -K_0 -K_1\\ \end{matrix} \right] [0−K0E−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\| ^2regularization\,\,term\\ subj.to\,\,: Constraints\,\,on\,\,u\,\,\\ \end{cases} ⎩ ⎨ ⎧minu ay−J˙a(q)q˙−Ja(q)u 2regularizationtermsubj.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 qu\varDelta t\leqslant \dot{q}_{\max}\\ \end{cases} {Acc:q¨min⩽u⩽q¨maxVel:q˙min⩽quΔt⩽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) τM(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)∑JiT[Ii]6×6Ji∈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∑k21(∂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 uτ 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} uq¨ 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 aqq¨K1y~˙K0y~⇒q~¨K1q~˙K0q~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 ) uM\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) uM(q)(q¨dK1q~˙K0q~)C(q,q˙)q˙g(q) 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 ) yf\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) yf(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] τuτ,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} ayy¨dK1y~˙K0y~
Inner loop : sekect torque control u τ u\tau uτ 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}} Jais 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 uMa_{\mathrm{q}}C\dot{q}g uMaqCq˙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 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˙−JaM−1(u−Cq˙−g) 2subj.to:u−⩽u⩽u optimization variable u ∈ R n u\in \mathbb{R} ^n u∈Rn
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}gu\\ u_-\leqslant u\in \mathbb{R} ^n\,\,\leqslant u_\,\,\\ \end{array}\\ \end{cases} ⎩ ⎨ ⎧u,q¨min ay−J˙aq˙−Jaq¨ 2subj.to:Mq¨Cq˙guu−⩽u∈Rn⩽u optimization variable u , q ¨ ∈ R n u,\ddot{q}\in \mathbb{R} ^n u,q¨∈Rn
