阻抗与导纳控制Impedance & Admittance Control

1为什么机器人需要它Why robots need this

2直觉：手爪应该"像什么"Intuition: what should the gripper feel like?

• 调节阻尼 \(d\)，让阻尼比 \(\zeta\) 落在 0.9 到 1.1 之间，然后"敲一下"或者拖开方块再松手：它回来时不再振荡。对默认的 \(m_d=2\) kg、\(k=250\) N/m，需要多大的 \(d\)？Adjust the damping \(d\) until the damping ratio \(\zeta\) is between 0.9 and 1.1, then "Tap" or drag the block away and let go: it comes back without ringing. For the default \(m_d=2\) kg and \(k=250\) N/m, what \(d\) does that take?
• 打开"持续推 20 N"，调节刚度，让方块静止时的变形正好是 5 cm（误差 ±5 mm，等它停稳）。你选的 \(k\) 是多少？\(m_d\) 和 \(d\) 对这个变形有影响吗？Switch on "Hold 20 N" and adjust the stiffness so that the block settles at a deflection of exactly 5 cm (within ±5 mm, wait until it stops). Which \(k\) did you pick? Do \(m_d\) and \(d\) affect this deflection?

3动手玩：同一个目标，两种实现Play: one target, two implementations

• 阻抗模式，表面选"钢"，把 \(x_0\) 拖进墙里至少 3 cm：接触力平稳下来（最近 0.5 s 的峰峰值小于 1 N），并且等于 \(k(x_0-x_w)\)，误差不超过摩擦力 \(f_c\) 加 1 N。墙比控制器硬一千倍，为什么没有问题？Impedance mode, surface "steel", drag \(x_0\) at least 3 cm into the wall: the contact force settles (peak-to-peak below 1 N over the last 0.5 s) at \(k(x_0-x_w)\), within the friction force \(f_c\) plus 1 N. The wall is a thousand times stiffer than the controller; why is that no problem?
• 切换到导纳模式，保持钢墙（或木头），让手爪顶在墙上：它开始剧烈"打颤"（最近 0.5 s 接触力峰峰值超过 100 N）。再换成泡沫试试。Switch to admittance mode, keep the steel (or wood) wall and push the gripper against it: it starts to "chatter" violently (contact force peak-to-peak above 100 N over the last 0.5 s). Then try foam.
• 把摩擦 \(f_c\) 调到 15 N 以上、刚度 \(k\le600\) N/m，在空中（不碰墙）把 \(x_0\) 拖动几厘米以上，等手爪停稳，分别在两种模式下看位置误差 \(|x-x_0|\)：阻抗模式超过 1 cm，导纳模式小于 1 mm。Set the friction \(f_c\) above 15 N and the stiffness \(k\le600\) N/m, move \(x_0\) by more than a few centimetres in free space (away from the wall), let the gripper stop, and compare the position error \(|x-x_0|\) in the two modes: above 1 cm in impedance mode, below 1 mm in admittance mode.

4第二、第三视角：接触稳定性与能量Second and third views: contact stability and energy

稳定性地图：谁在什么环境里会失稳A stability map: who goes unstable against what

• 导纳地图：把标记放到钢附近（\(k_e\ge10^6\) N/m），找出能让它稳定的最小虚拟质量（标记必须在稳定区，且 \(m_a\le60\) kg）。这么重的虚拟质量，在空中被人推的时候会是什么感觉？Admittance map: put the marker near steel (\(k_e\ge10^6\) N/m) and find the smallest virtual mass that is stable there (the marker must be in the stable region with \(m_a\le60\) kg). How would such a heavy virtual mass feel when a person pushes it in free space?
• 阻抗地图：把延迟调到 3 ms 或更多，在软环境（\(k_e\le10^4\) N/m）里找一个不稳定的点，要求渲染刚度 \(k\le3\times10^5\) N/m。阻抗控制的稳定边界主要取决于哪个轴？Impedance map: set the delay to 3 ms or more and find an unstable point in a soft environment (\(k_e\le10^4\) N/m) with rendered stiffness \(k\le3\times10^5\) N/m. Which axis mainly decides impedance control's stability boundary?

能量视角：无源性，以及为什么"变刚度"会出事The energy view: passivity, and why "varying the stiffness" can go wrong

• 关掉能量罐，调节调制频率 \(\Omega/\omega_n\)（保持 \(a=0.5\)、\(\zeta=0.05\)），让系统能量增长到初始值 0.5 J 的 10 倍以上。哪个频率比最危险？\(\Omega=\omega_n\) 时会怎样？With the tank off, adjust the modulation frequency \(\Omega/\omega_n\) (keep \(a=0.5\), \(\zeta=0.05\)) until the system energy grows to more than 10 times its initial 0.5 J. Which frequency ratio is the most dangerous? What happens at \(\Omega=\omega_n\)?
• 保持危险的调制（\(a\ge0.4\)，\(\Omega/\omega_n\) 在 1.8 到 2.2 之间），打开能量罐并按"复位"，让它运行 10 秒以上：系统能量始终不超过 1.05 J。刚度曲线发生了什么变化？Keep a dangerous modulation (\(a\ge0.4\), \(\Omega/\omega_n\) between 1.8 and 2.2), switch the tank on, press "Reset" and let it run for more than 10 seconds: the system energy never exceeds 1.05 J. What happened to the stiffness curve?

5再看数学The math

提示：把鼠标悬停在公式中带底色的符号上，对应的图元素会被高亮：\(m_d,d,k,f_{\mathrm{ext}}\) 对应 FIG 1 的方块、阻尼器、弹簧和手的力箭头；\(k_e\) 对应 FIG 2 的墙，\(x_c\) 对应 FIG 2 的参考位置空心框；\(\rho\) 对应 FIG 3 的稳定边界；\(\dot k\) 项对应 FIG 4 的紫色功率曲线，\(T\) 对应能量罐；\(K\) 对应 FIG 5 的变形椭圆，\(J^\top\) 对应 FIG 5 关节处的力矩弧；\(S\) 对应 FIG 6 的选择坐标轴。Tip: hover the highlighted symbols in the formulas and the matching figure element lights up: \(m_d,d,k,f_{\mathrm{ext}}\) are FIG 1's block, damper, spring and hand-force arrow; \(k_e\) is FIG 2's wall and \(x_c\) its hollow reference frame; \(\rho\) is FIG 3's stability boundary; the \(\dot k\) term is FIG 4's purple power curve and \(T\) its energy tank; \(K\) is FIG 5's deflection ellipse and \(J^\top\) the torque arcs at its joints; \(S\) is FIG 6's selection axes.

目标阻抗与因果关系The target impedance and causality

在 \(n\) 维任务空间里（平面位置 \(n=2\)，空间位姿 \(n=6\)），目标行为写成

In an \(n\)-dimensional task space (\(n=2\) for planar position, \(n=6\) for a spatial pose) the target behaviour is

其中 \(M_d,D,K\) 是对称正定矩阵，\(\mathbf F_{\mathrm{ext}}\) 是外界作用在手爪上的力（7.6 节的力旋量去掉力矩部分）。\(Z(s)\) 是从速度到力的传递函数（\(\mathbf F=Z\mathbf V\)，\(\mathbf V=s\tilde{\mathbf X}\)）。Hogan 的论点是关于因果的：环境（比如一面墙）本身往往是一个导纳（给它力，它决定位置变化很小），两个"导纳"不能串在一起，所以机器人应该表现为阻抗，"你动我，我给力"。阻抗控制和导纳控制的区别不在目标，而在实现的因果方向：

where \(M_d,D,K\) are symmetric positive definite and \(\mathbf F_{\mathrm{ext}}\) is the force the outside world applies to the gripper (the wrench of Lesson 7.6 without its moment). \(Z(s)\) is the transfer function from velocity to force (\(\mathbf F=Z\mathbf V\), \(\mathbf V=s\tilde{\mathbf X}\)). Hogan's argument is about causality: the environment (a wall, say) is often itself an admittance (push it and it decides how little it moves), and two "admittances" cannot be connected in series, so the robot should present an impedance, "you move me, I give force". Impedance and admittance control differ not in the target but in the causal direction of the implementation:

机械臂上的阻抗控制Impedance control on an arm

对力矩控制的机械臂 \(M(\mathbf q)\ddot{\mathbf q}+C(\mathbf q,\dot{\mathbf q})\dot{\mathbf q}+\mathbf g(\mathbf q)=\boldsymbol\tau+J^\top\mathbf F_{\mathrm{ext}}\)（8.3 节；外力经 7.6 节的 \(J^\top\) 变成关节力矩），最常用的笛卡尔阻抗控制律是

For a torque-controlled arm \(M(\mathbf q)\ddot{\mathbf q}+C(\mathbf q,\dot{\mathbf q})\dot{\mathbf q}+\mathbf g(\mathbf q)=\boldsymbol\tau+J^\top\mathbf F_{\mathrm{ext}}\) (Lesson 8.3; the external force enters through the \(J^\top\) of Lesson 7.6), the most common Cartesian impedance law is

它只用到正运动学、雅可比和重力模型，不需要力传感器。代入动力学并利用 8.3 节的操作空间形式（\(J\) 可逆时），闭环是

It needs only forward kinematics, the Jacobian and a gravity model, and no force sensor. Substituting into the dynamics and using the operational-space form of Lesson 8.3 (for invertible \(J\)), the closed loop is

刚度和阻尼正是我们想要的，但质量是手臂自己的等效质量 \(\Lambda(\mathbf q)\)（8.3 节 FIG 3 的椭圆），随姿态变化，还多了速度项 \(\boldsymbol\mu\)。要把质量也改成任意的 \(M_d\)（"惯量整形"），控制律里必须出现 \(\Lambda M_d^{-1}\mathbf F_{\mathrm{ext}}\)，也就是必须测量外力，并用 \(\Lambda\) 和 \(\boldsymbol\mu\) 做补偿：这正是 13.6 节操作空间控制的内容。实践中大多数力矩控制臂选择保留 \(\Lambda\)：不用力传感器，鲁棒，而且下面的能量论证完全成立。

The stiffness and damping are what we asked for, but the mass is the arm's own apparent mass \(\Lambda(\mathbf q)\) (the ellipse of Lesson 8.3's FIG 3), which changes with posture, plus a velocity term \(\boldsymbol\mu\). Changing the mass to an arbitrary \(M_d\) as well ("inertia shaping") requires a term \(\Lambda M_d^{-1}\mathbf F_{\mathrm{ext}}\) in the control law, i.e. the external force must be measured, and \(\Lambda\) and \(\boldsymbol\mu\) must be compensated: that is the operational-space control of Lesson 13.6. In practice most torque-controlled arms keep \(\Lambda\): no force sensor, robust, and the energy argument below holds exactly.

关节空间版本。8.3 节的"PD + 重力补偿" \(\boldsymbol\tau=\mathbf g+K_q(\mathbf q_0-\mathbf q)-D_q\dot{\mathbf q}\) 就是一个关节阻抗控制器。对小位移 \(\delta\mathbf x=J\delta\mathbf q\)，笛卡尔刚度 \(K\) 对应的关节刚度是 \(K_q=J^\top KJ\)（外力不为零时还要加上 \(\partial J^\top/\partial\mathbf q\) 与力的乘积项）。反过来，同样的关节刚度在不同姿态下对应不同的笛卡尔刚度，这就是为什么要在任务空间里规定 \(K\)。

Joint-space version. Lesson 8.3's "PD + gravity compensation", \(\boldsymbol\tau=\mathbf g+K_q(\mathbf q_0-\mathbf q)-D_q\dot{\mathbf q}\), is a joint impedance controller. For small displacements \(\delta\mathbf x=J\delta\mathbf q\), a Cartesian stiffness \(K\) corresponds to the joint stiffness \(K_q=J^\top KJ\) (with a non-zero external force, add a term with \(\partial J^\top/\partial\mathbf q\) times that force). Conversely, the same joint stiffness means a different Cartesian stiffness in every posture, which is why \(K\) is specified in task space.

能量函数与无源性The energy function and passivity

取 \(W=\tfrac12\dot{\mathbf q}^\top M\dot{\mathbf q}+\tfrac12\tilde{\mathbf x}^\top K\tilde{\mathbf x}\)（动能加虚拟弹簧的势能）。8.3 节的功率平衡（\(\dot M-2C\) 斜对称）给出 \(\tfrac{d}{dt}\tfrac12\dot{\mathbf q}^\top M\dot{\mathbf q}=\dot{\mathbf q}^\top(\boldsymbol\tau+J^\top\mathbf F_{\mathrm{ext}}-\mathbf g)\)。代入控制律，用 \(\dot{\mathbf x}=J\dot{\mathbf q}\)：

Take \(W=\tfrac12\dot{\mathbf q}^\top M\dot{\mathbf q}+\tfrac12\tilde{\mathbf x}^\top K\tilde{\mathbf x}\) (kinetic energy plus the virtual spring's potential energy). Lesson 8.3's power balance (\(\dot M-2C\) skew-symmetric) gives \(\tfrac{d}{dt}\tfrac12\dot{\mathbf q}^\top M\dot{\mathbf q}=\dot{\mathbf q}^\top(\boldsymbol\tau+J^\top\mathbf F_{\mathrm{ext}}-\mathbf g)\). Substitute the control law and use \(\dot{\mathbf x}=J\dot{\mathbf q}\):

两个结论。(1) 空中（\(\mathbf F_{\mathrm{ext}}=0\)）：\(\dot W=-\dot{\mathbf x}^\top D\dot{\mathbf x}\le0\)，\(W\) 是 13.2 节意义下的 Lyapunov 函数；\(\dot W=0\) 要求 \(\dot{\mathbf x}=0\)，再由 LaSalle 原理得 \(\mathbf x\to\mathbf x_0\)（\(J\) 非奇异时）。(2) 对任意外力，积分得 \(\int_0^t\dot{\mathbf x}^\top\mathbf F_{\mathrm{ext}}\,dt\ge W(t)-W(0)\ge-W(0)\)：外界从机器人那里最多能取走它初始储存的能量，这就是无源性的定义。墙、弹簧、阻尼器、人手的被动部分都是无源的；两个无源系统的反馈连接仍然无源，总能量有界，所以接触稳定（Colgate 与 Hogan 1988 的耦合稳定性）。

Two conclusions. (1) In free space (\(\mathbf F_{\mathrm{ext}}=0\)): \(\dot W=-\dot{\mathbf x}^\top D\dot{\mathbf x}\le0\), so \(W\) is a Lyapunov function in the sense of Lesson 13.2; \(\dot W=0\) forces \(\dot{\mathbf x}=0\), and LaSalle's principle gives \(\mathbf x\to\mathbf x_0\) (for non-singular \(J\)). (2) For any external force, integrating gives \(\int_0^t\dot{\mathbf x}^\top\mathbf F_{\mathrm{ext}}\,dt\ge W(t)-W(0)\ge-W(0)\): the outside world can extract at most the energy initially stored in the robot, which is the definition of passivity. Walls, springs, dampers and the passive part of a human hand are passive; a feedback connection of two passive systems is passive again, the total energy stays bounded, and the contact is stable (the coupled stability of Colgate and Hogan, 1988).

这个证明隐含了两个理想化：控制律是连续时间的，电机力矩如实到达关节。采样和延迟会让虚拟弹簧"晚一步"出力，相当于一个负阻尼。对触觉设备，Colgate 与 Schenkel（1997）给出了采样渲染的无源性条件 \(b>\tfrac{KT}{2}+|B|\)：要以采样周期 \(T\) 渲染刚度 \(K\) 和虚拟阻尼 \(B\)，设备本身必须有足够的物理阻尼 \(b\)。这就是 FIG 3 阻抗地图上方的不稳定区。

The proof hides two idealizations: the control law is continuous-time, and the motor torque arrives at the joint exactly. Sampling and delay make the virtual spring push "one step late", which acts like negative damping. For haptic devices, Colgate and Schenkel (1997) derived the passivity condition of sampled rendering, \(b>\tfrac{KT}{2}+|B|\): to render a stiffness \(K\) and virtual damping \(B\) at sampling period \(T\), the device itself needs enough physical damping \(b\). That is the unstable region at the top of FIG 3's impedance map.

导纳控制与它的环路增益Admittance control and its loop gain

导纳控制器在每个周期 \(T\)（1 ms）里用测得的力 \(\mathbf f\) 积分一次虚拟动力学（半隐式欧拉，2.5 节），把结果交给位置伺服：

Every period \(T\) (1 ms) the admittance controller integrates the virtual dynamics once with the measured force \(\mathbf f\) (semi-implicit Euler, Lesson 2.5) and hands the result to the position servo:

在一维里，环境是刚度 \(\hl{ke}{k_e}\) 的弹簧，伺服把参考 \(x_c\) 变成实际位置 \(x\) 的传递函数记为 \(G_s(s)\)（带宽 \(\omega_s\)）。力 \(f=-k_e(x-x_w)\) 又回到导纳模型，形成一个闭环，其环路增益为

In one dimension the environment is a spring of stiffness \(\hl{ke}{k_e}\), and the servo turns the reference \(x_c\) into the actual position \(x\) through a transfer function \(G_s(s)\) (bandwidth \(\omega_s\)). The force \(f=-k_e(x-x_w)\) feeds back into the admittance model, closing a loop with loop gain

（\(k_e\) 远大于伺服刚度 \(k_s\) 时，墙把机器人挡住，\(x\) 几乎不随 \(x_c\) 动，于是有效增益饱和为 \(k_s\)，即串联刚度 \(k_{\mathrm{eff}}\)。）（近似式里忽略了伺服 \(G_s\) 的相位。）增益在 \(\omega_c\approx\sqrt{k_{\mathrm{eff}}/m_a}\) 处降到 1。虚拟质量–弹簧本身已经贡献了接近 \(-180^\circ\) 的相位，只要伺服和延迟在 \(\omega_c\) 处再滞后一点，相位裕度就没了：所以 \(\omega_c\) 必须明显低于 \(\omega_s\)。这给出三条补救：加大 \(m_a\)（代价是空中推起来很沉），加大 \(d_a\)（代价是很黏），或者降低伺服刚度（代价是空中精度）。没有免费的午餐：导纳控制在硬接触里稳定，本质上就是让机器人在那里变得又重又慢。

(When \(k_e\) far exceeds the servo stiffness \(k_s\), the wall holds the robot back and \(x\) hardly follows \(x_c\), so the effective gain saturates at \(k_s\): the series stiffness \(k_{\mathrm{eff}}\).) (The approximation ignores the phase of the servo \(G_s\).) The gain falls to 1 near \(\omega_c\approx\sqrt{k_{\mathrm{eff}}/m_a}\). The virtual mass–spring already contributes close to \(-180^\circ\) of phase, so a little extra lag from the servo and the delay at \(\omega_c\) wipes out the phase margin: \(\omega_c\) must stay clearly below \(\omega_s\). That gives three remedies: raise \(m_a\) (it then feels heavy to push in free space), raise \(d_a\) (it feels sticky), or lower the servo stiffness (it loses free-space accuracy). There is no free lunch: making admittance control stable in stiff contact essentially means making the robot heavy and slow there.

阻抗与导纳的对偶可以说得更精确：阻抗控制在渲染低刚度、接触硬环境时好；导纳控制在渲染高刚度、接触软环境（或在空中）时好。Ott、Mukherjee 与 Nakamura（2010）提出把两者按比例混合，在两种情形之间切换或插值。

The duality can be stated more precisely: impedance control is good at rendering low stiffness against stiff environments; admittance control is good at rendering high stiffness against soft environments (or in free space). Ott, Mukherjee and Nakamura (2010) proposed blending the two in proportion, switching or interpolating between the two regimes.

怎样选 \(K\) 和 \(D\)Choosing \(K\) and \(D\)

刚度按任务和不确定性选。在需要精度的方向上硬，在位置不确定、需要接触的方向上软。外力 \(\mathbf F\) 下的静态变形是 \(\Delta\mathbf x=K^{-1}\mathbf F\)：如果表面高度的不确定性是 \(\pm\delta\)，想把接触力的波动压在 \(\pm\Delta f\) 以内，法向刚度就要 \(k_n\lesssim\Delta f/\delta\)。刚度矩阵不必是对角的：\(K=R\,\mathrm{diag}(k_1,k_2)R^\top\)，主轴可以对准表面法向。注意：除非力恰好沿着 \(K\) 的主轴，变形方向和力的方向不一致（FIG 5）。

Pick the stiffness from the task and the uncertainty. Stiff in directions that need accuracy, soft in directions where the position is uncertain and contact is expected. The static deflection under a force \(\mathbf F\) is \(\Delta\mathbf x=K^{-1}\mathbf F\): if the surface height is uncertain by \(\pm\delta\) and the contact force must vary by at most \(\pm\Delta f\), the normal stiffness must satisfy \(k_n\lesssim\Delta f/\delta\). The stiffness matrix need not be diagonal: \(K=R\,\mathrm{diag}(k_1,k_2)R^\top\) can align its principal axes with the surface normal. Note that unless the force lies exactly along a principal axis of \(K\), the deflection points in a different direction than the force (FIG 5).

阻尼按质量选。一维时 \(d=2\zeta\sqrt{k\,m}\)，\(m\) 是实际的质量（阻抗控制里是 \(\Lambda\)）。如果固定 \(d\) 而改变 \(k\) 或姿态，阻尼比就会跟着漂。多维时一个常用的做法是"双对角化"（Albu-Schäffer 等，2003）：

Pick the damping from the mass. In 1D, \(d=2\zeta\sqrt{k\,m}\) with \(m\) the actual mass (\(\Lambda\) under impedance control). Keep \(d\) fixed while \(k\) or the posture changes and the damping ratio drifts. In several dimensions a common recipe is "double diagonalization" (Albu-Schäffer et al., 2003):

它让 \(\Lambda\ddot{\mathbf x}+D\dot{\mathbf x}+K\mathbf x=0\) 的每一个模态都有阻尼比 \(\zeta\)（把 \(\mathbf y=\Lambda^{1/2}\mathbf x\) 代入即可验证）。FIG 5 就用这个公式，每 1 ms 按当前姿态重算一次。转动部分的刚度单位是 N·m/rad，作用在姿态误差 \(\Log(R_0^\top R)\)（5.6 节）上；平移和转动的刚度不能直接比较，6 维 \(K\) 通常写成分块对角。

It gives every mode of \(\Lambda\ddot{\mathbf x}+D\dot{\mathbf x}+K\mathbf x=0\) the damping ratio \(\zeta\) (substitute \(\mathbf y=\Lambda^{1/2}\mathbf x\) to check). FIG 5 uses exactly this formula, recomputed from the current posture every 1 ms. The rotational stiffness has units N·m/rad and acts on the orientation error \(\Log(R_0^\top R)\) (Lesson 5.6); translational and rotational stiffnesses cannot be compared directly, so a 6D \(K\) is usually block-diagonal.

变阻抗与能量罐Variable impedance and the energy tank

让 \(K\) 随时间变，重复上面的推导，\(\tfrac{d}{dt}\tfrac12\tilde{\mathbf x}^\top K\tilde{\mathbf x}\) 多出 \(\tfrac12\tilde{\mathbf x}^\top\dot K\tilde{\mathbf x}\)：

Let \(K\) vary with time and repeat the derivation above: \(\tfrac{d}{dt}\tfrac12\tilde{\mathbf x}^\top K\tilde{\mathbf x}\) gains \(\tfrac12\tilde{\mathbf x}^\top\dot K\tilde{\mathbf x}\):

罐子里的能量 \(T\) 低于下限 \(T_{\min}\) 时，会注入能量的刚度变化被暂停（\(\dot K\) 置零）。没有罐子时，\(\dot K\) 项可以是正的，无源性就丢了；FIG 4 里 \(a=0.5\)、\(\Omega=2\omega_n\) 时，能量在 10 秒内从 0.5 J 涨到约 \(10^6\) J。有罐子时 \(\tfrac{d}{dt}(W+T)=\dot{\mathbf x}^\top\mathbf F_{\mathrm{ext}}\)，与固定刚度时一样无源：罐子只允许把以前耗散掉的能量"借回来"改变刚度。（FIG 4 还给罐子设了 1 J 的上限，防止长期积攒的能量一次性释放。）

When the tank energy \(T\) falls below a floor \(T_{\min}\), stiffness changes that would inject energy are paused (\(\dot K\) set to zero). Without a tank the \(\dot K\) term can be positive and passivity is lost: in FIG 4 with \(a=0.5\) and \(\Omega=2\omega_n\) the energy grows from 0.5 J to about \(10^6\) J in 10 seconds. With the tank, \(\tfrac{d}{dt}(W+T)=\dot{\mathbf x}^\top\mathbf F_{\mathrm{ext}}\), just as passive as with constant stiffness: the tank only lets the controller "borrow back" energy that was dissipated earlier to change the stiffness. (FIG 4 also caps the tank at 1 J so that energy saved up over a long time cannot be released all at once.)

力/位混合控制（简述）Hybrid force/position control (briefly)

另一条思路来自 Mason（1981）：接触几何把任务空间分成两组方向。沿表面法向，环境决定了位置（自然约束：不能穿过去），我们只能规定力；沿表面切向，环境决定了力（无摩擦时为零），我们只能规定位置。Raibert 与 Craig（1981）据此用一个对角的选择矩阵 \(S\)（1 = 位置控制，0 = 力控制）在任务坐标系里把两个控制器拼起来：

Another line of thought comes from Mason (1981): the contact geometry splits task space into two sets of directions. Along the surface normal the environment determines the position (a natural constraint: you cannot go through), so we can only prescribe force; along the surface the environment determines the force (zero without friction), so we can only prescribe position. Raibert and Craig (1981) used a diagonal selection matrix \(S\) (1 = position-controlled, 0 = force-controlled) in the task frame to combine two controllers:

它能把力精确调到 \(\mathbf f_d\)，与表面在哪无关；阻抗控制只能间接地通过 \(K(\mathbf x_0-\mathbf x)\) 得到力，表面高度差 \(\delta\) 就带来 \(k_n\delta\) 的力误差（FIG 6）。代价是：\(S\) 要求事先知道约束方向；力控制方向上没有位置参考，一旦失去接触就会加速；在接触刚度很高时，力回路的增益必须很小才能稳定（又是 FIG 3 的环路增益问题）。所以在工业里常见的是两者结合：用阻抗控制兜底的稳定性，在法向加一个慢的力积分环。

It regulates the force exactly to \(\mathbf f_d\), wherever the surface is; impedance control only obtains force indirectly through \(K(\mathbf x_0-\mathbf x)\), so a surface height error \(\delta\) becomes a force error \(k_n\delta\) (FIG 6). The price: \(S\) requires knowing the constraint directions in advance; the force-controlled direction has no position reference and accelerates once contact is lost; and against high contact stiffness the force loop's gain must be small to stay stable (FIG 3's loop-gain problem again). That is why industry often combines the two: impedance control for baseline stability, plus a slow force integrator along the normal.

6机器人中的它：笛卡尔阻抗与擦桌子In robotics: Cartesian impedance and wiping a table

• 把刚度调成各向异性（\(k_1/k_2\ge4\) 或 \(k_2/k_1\ge4\)），用摇杆施加至少 10 N 的力，让手爪停稳后，变形方向与力的方向相差超过 25°。什么方向的力不会产生这种偏差？Make the stiffness anisotropic (\(k_1/k_2\ge4\) or \(k_2/k_1\ge4\)), apply at least 10 N with the joystick, and once the gripper is at rest, get the deflection to point more than 25° away from the force. Which force directions produce no such deviation?
• 打开波浪桌面和"擦拭"，调节竖直方向的刚度和 \(\mathbf x_0\) 的高度，让一个完整的往返（6 s）里接触力始终在 4 N 到 16 N 之间。为什么法向刚度要低？试试把它调到 1000 N/m 以上。Switch on the wavy table and "Wipe", then tune the vertical stiffness and the height of \(\mathbf x_0\) so that the contact force stays between 4 N and 16 N for a whole back-and-forth stroke (6 s). Why must the normal stiffness be low? Try raising it above 1000 N/m.

• 保持倾斜 \(|\delta|\le1^\circ\)，把表面抬高 \(\Delta h\ge1.5\) cm，等一个完整往返：混合控制的力始终在 \(10\pm0.5\) N 以内，而阻抗控制的平均力偏离 10 N 超过 4 N。再把倾斜调到 10°，两者各自怎样？Keep the tilt \(|\delta|\le1^\circ\), raise the surface by \(\Delta h\ge1.5\) cm and wait for a full stroke: the hybrid force stays within \(10\pm0.5\) N, while the impedance controller's mean force is off from 10 N by more than 4 N. Then tilt the surface by 10°: how does each one fare?

7常见误区Common pitfalls

8小练习Check yourself

9延伸阅读Further reading

• N. Hogan, Impedance Control: An Approach to Manipulation, Parts I–III, ASME J. Dynamic Systems, Measurement, and Control 107(1), 1985 — 阻抗控制的奠基论文：因果、能量与操作The founding papers of impedance control: causality, energy and manipulation
• M. T. Mason, Compliance and Force Control for Computer Controlled Manipulators, IEEE Trans. Systems, Man, and Cybernetics 11(6), 1981 — 自然约束与人工约束Natural and artificial constraints
• M. H. Raibert & J. J. Craig, Hybrid Position/Force Control of Manipulators, ASME J. Dynamic Systems, Measurement, and Control 103(2), 1981 — 选择矩阵与力/位混合控制The selection matrix and hybrid position/force control
• J. E. Colgate & N. Hogan, Robust Control of Dynamically Interacting Systems, Int. J. Control 48(1), 1988 — 无源性与耦合稳定性Passivity and coupled stability
• J. E. Colgate & G. G. Schenkel, Passivity of a Class of Sampled-Data Systems: Application to Haptic Interfaces, J. Robotic Systems 14(1), 1997 — 采样渲染的无源性条件 \(b>KT/2+|B|\)The passivity condition for sampled rendering, \(b>KT/2+|B|\)
• D. E. Whitney, Historical Perspective and State of the Art in Robot Force Control, Int. J. Robotics Research 6(1), 1987 — 力控制早期方法的综述A survey of the early force-control methods
• A. Albu-Schäffer, C. Ott, U. Frese & G. Hirzinger, Cartesian Impedance Control of Redundant Robots: Recent Results with the DLR-Light-Weight-Arms, ICRA 2003 — 力矩控制臂上的笛卡尔阻抗与双对角化阻尼设计Cartesian impedance on torque-controlled arms and the double-diagonalization damping design
• C. Ott, R. Mukherjee & Y. Nakamura, Unified Impedance and Admittance Control, ICRA 2010 — 阻抗与导纳的对偶及其融合The duality of impedance and admittance, and how to blend them
• F. Ferraguti, C. Secchi & C. Fantuzzi, A Tank-Based Approach to Impedance Control with Variable Stiffness, ICRA 2013 — 用能量罐保持变刚度的无源性Keeping variable stiffness passive with an energy tank
• B. Siciliano, L. Sciavicco, L. Villani & G. Oriolo, Robotics: Modelling, Planning and Control, Springer, 2009, Ch. 9 — 力控制：阻抗、导纳、力/位混合与并联控制Force control: impedance, admittance, hybrid and parallel force/position control
• K. Lynch & F. Park, Modern Robotics, Ch. 11 — 力控制、混合运动–力控制与阻抗控制Force control, hybrid motion–force control and impedance control
• R. Martín-Martín, M. A. Lee, R. Gardner, S. Savarese, J. Bohg & A. Garg, Variable Impedance Control in End-Effector Space: An Action Space for Reinforcement Learning in Contact-Rich Tasks, IROS 2019 — 把阻抗控制器当作学习的动作空间The impedance controller as an action space for learning