足式与人形机器人基础Legged & Humanoid Basics

轮子需要一条路，腿只需要一个个落脚点：台阶、碎石、楼梯、灾后废墟，只有足式机器人进得去；人形机器人更是希望直接使用为人类设计的世界。但是腿把机械臂里没有的三个难题带了进来：

• 身体是"浮动"的。机械臂的底座拧在桌上，关节一转末端就动。足式机器人的躯干（浮动基座）没有被任何东西固定，它的位姿属于 \(SE(3)\)（7.2 节），其中 6 个自由度没有电机：关节力矩只能通过脚底的接触力，间接地推动躯干。
• 接触是间歇的、单向的。脚只能推地，不能拉地（法向力 \(\ge 0\)）；切向力还不能超过摩擦锥（8.4 节）。每迈一步，脚离地、落地，接触模式就切换一次。
• 平衡不是目标，是约束。静止的机械臂不会倒；站着的人形机器人像竖在指尖的一支笔，没有控制就在零点几秒内倒下。

这一节要建立一套最小的"足式直觉工具箱"：支撑多边形（什么时候不用动也不会倒）、浮动基座上的力与力矩平衡（脚到底能给躯干什么）、线性倒立摆与 ZMP / 捕获点（怎样走、怎样被推了还不倒）、步态（四足怎样安排脚的节奏）。真实的足式控制器（Part IV 的 MPC 与全身控制，Part VI 的强化学习）都建立在这些概念之上。

A wheel needs a road; a leg needs only a sequence of footholds: stairs, rubble, rocks, disaster sites are reachable only by legged robots, and humanoids would let robots use a world built for people. But legs bring three difficulties that a manipulator never had:

• The body is "floating". An arm's base is bolted to the table, so turning a joint moves the tip. A legged robot's trunk (the floating base) is held by nothing; its pose lives in \(SE(3)\) (Lesson 7.2), and 6 of its degrees of freedom have no motor: joint torques can only move the trunk indirectly, through contact forces at the feet.
• Contact is intermittent and one-sided. A foot can push on the ground but not pull (normal force \(\ge 0\)), and its tangential force must stay inside the friction cone (Lesson 8.4). Every step lifts a foot and puts it down, switching the contact mode.
• Balance is a constraint, not a goal. A resting arm does not fall over; a standing humanoid is like a pencil balanced on a fingertip and topples within a fraction of a second without control.

This lesson builds a minimal legged-locomotion toolbox: the support polygon (when does a robot not fall even if it does nothing?), force and moment balance on a floating base (what can the feet actually give the trunk?), the linear inverted pendulum with ZMP and the capture point (how to walk, and how to survive a push), and gaits (how four legs share the rhythm). Real legged controllers (MPC and whole-body control in Part IV, reinforcement learning in Part VI) are built on these ideas.

把机器人摆在地上，所有脚底接触点的凸包叫支撑多边形。只要质心（COM）沿重力方向在地面上的投影落在支撑多边形内，机器人静止就不会倒；投影越过某条边，它就绕这条边翻倒。投影点到最近边的距离叫稳定裕度。四足站立时多边形很大；抬起一条腿，它变成三角形，质心必须先挪进三角形；双足时多边形只是两只脚和它们之间的区域。

斜坡会"移动"投影点：质心高度为 \(h\)、坡度为 \(\alpha\) 时，重力方向的投影相对于"沿坡面法向"的投影向下坡方向偏移 \(h\tan\alpha\)，所以重心越高，在斜坡上越容易翻。同时，沿坡面下滑的趋势 \(\tan\alpha\) 超过摩擦系数 \(\mu\) 时，脚会打滑，这时翻倒之前先滑倒（8.4 节）。

Put the robot on the ground. The convex hull of all foot contact points is the support polygon. As long as the ground projection of the center of mass (COM) along gravity lies inside it, the robot stays up at rest; once the projection crosses an edge, the robot rotates about that edge and tips. The distance from the projection to the nearest edge is the stability margin. A standing quadruped has a large polygon; lifting a leg turns it into a triangle and the COM must first move into the triangle; for a biped the polygon is just the two feet and the strip between them.

A slope shifts the projection: with the COM at height \(h\) over a slope of angle \(\alpha\), the projection along gravity is displaced downhill by \(h\tan\alpha\) relative to the projection along the surface normal, so the higher the COM, the sooner the robot tips on a slope. Meanwhile the downhill pull \(\tan\alpha\) exceeding the friction coefficient \(\mu\) makes the feet slide, so the robot slips before it tips (Lesson 8.4).

支撑多边形只回答"静止时倒不倒"。要回答"能不能走起来、跑起来"，必须问：脚能给躯干什么力？每只脚给出一个地面反力 \(\mathbf f_i\)，它受两条约束：法向分量 \(f_z\ge0\)（单向），切向分量 \(|f_x|\le\mu f_z\)（摩擦锥，半角 \(\arctan\mu\)，8.4 节）。把所有脚的力加起来，再加上重力，就是整个机器人的牛顿–欧拉方程（8.2 节）：

The support polygon only answers "does it fall when standing still?". To ask whether the robot can walk or run, we must ask what the feet can give the trunk. Each foot supplies a ground reaction force \(\mathbf f_i\) with two constraints: normal component \(f_z\ge0\) (one-sided) and tangential component \(|f_x|\le\mu f_z\) (the friction cone with half-angle \(\arctan\mu\), Lesson 8.4). Add up all the foot forces plus gravity and you have the Newton–Euler equations of the whole robot (Lesson 8.2):

左边是整台机器人质心的加速度和绕质心的角动量变化率（所有关节力矩都是内力，在求和中抵消了，所以式子里没有 \(\boldsymbol\tau\)）。右边只有脚上的力。这就是"躯干只能通过脚被推动"的数学形式。下图是矢状面（侧视图）里的两只脚：拖动脚尖的力手柄，力被自动投影到摩擦锥内（最近点投影，1.9 节）。紫色箭头是躯干受到的净力 \(m\ddot{\mathbf c}\)，琥珀色弧线是绕质心的净力矩。想让机器人静止，力要满足 \(\sum f_z=mg\)、\(\sum f_x=0\) 和力矩为零。

The left side is the acceleration of the whole robot's COM and the rate of change of its angular momentum about the COM (every joint torque is an internal force and cancels in the sum, so no \(\boldsymbol\tau\) appears). The right side contains only forces at the feet. That is the mathematical form of "the trunk can only be pushed through the feet". The figure shows two feet in the sagittal plane (a side view): drag the force handles at the foot tips and each force is automatically projected into the friction cone (nearest-point projection, Lesson 1.9). The purple arrow is the net force on the trunk, \(m\ddot{\mathbf c}\); the amber arc is the net moment about the COM. To stand still the forces must satisfy \(\sum f_z=mg\), \(\sum f_x=0\) and zero moment.

现在加上动力学。假设质心保持恒定高度 \(z_c\)、腿没有质量、不考虑绕质心的角动量变化，那么质心的水平运动满足一个线性方程，叫线性倒立摆模型（LIPM）：

Now add dynamics. Assume the COM keeps a constant height \(z_c\), the legs are massless, and the angular momentum about the COM does not change. Then the horizontal COM motion obeys a linear equation, the linear inverted pendulum model (LIPM):

\(p\) 是脚下的支撑点（即 ZMP，下文解释）。对人形机器人 \(z_c\approx0.85\) m，\(\omega\approx3.4\ \mathrm{s^{-1}}\)，时间常数 \(1/\omega\approx0.3\) s：质心偏离脚 \(x-p\)，会以这个时间常数指数地"飞离"。走路就是每隔一步换一个支撑点，让质心在落到下一只脚上方时刚好被接住。下图让一个 LIPM 行走器按这个规则连续迈步：每一步结束时，把新脚放在 \(p_{\text{new}}=\hl{dcm}{\xi_{\text{end}}}-b\) 处（\(\xi=x+\dot x/\omega\) 是下文的"捕获点"，\(b\) 由期望速度和步时决定，见"再看数学"）。拖动滑块改变质心高度、步时和期望速度。

\(p\) is the support point under the foot (the ZMP, explained below). For a humanoid, \(z_c\approx0.85\) m, \(\omega\approx3.4\ \mathrm{s^{-1}}\) and the time constant \(1/\omega\approx0.3\) s: a COM displaced by \(x-p\) from the foot "flies away" exponentially with that time constant. Walking means switching the support point every step so that the COM is caught just as it arrives over the next foot. The figure runs an LIPM walker with that rule: at the end of each step the new foot goes to \(p_{\text{new}}=\hl{dcm}{\xi_{\text{end}}}-b\) (\(\xi=x+\dot x/\omega\) is the "capture point" below, and \(b\) depends on the desired speed and step time; see "The math"). Drag the sliders to change the COM height, step time and desired speed.

站着的机器人被突然推了一把，质心得到速度 \(\dot x_0\)。捕获点（Pratt 等 2006，也叫发散分量 DCM）\(\xi=x+\dot x/\omega\) 回答："脚落在哪里，机器人才会恰好停在那只脚上方？"答案就是 \(p=\xi\)：因为 \(\dot\xi=\omega(\xi-p)\)，脚放在 \(\xi\) 处，\(\xi\) 就不再变化，\(x\) 以时间常数 \(1/\omega\) 收敛到 \(\xi\)。但迈步需要时间 \(t_s\)，这段时间里 \(\xi\) 仍在以 \(e^{\omega t_s}\) 增长，所以实际要落的位置是 \(\xi(t_s)=\xi_0e^{\omega t_s}\)。

脚能够到的范围有限（腿长、关节限位），于是出现捕获区域：推得不太狠时，只靠踝关节的力矩把 ZMP 移到脚尖就够了（"踝策略"，\(\xi_0\) 落在脚的范围内）；更大的推力要迈步（"迈步策略"）；再大，脚够不到 \(\xi(t_s)\)，就一定倒。这就是人被推时会下意识迈出一步的原因：脚太小，踝策略的容量只有几厘米。

A standing robot gets a sudden shove and the COM acquires velocity \(\dot x_0\). The capture point (Pratt et al. 2006; also the divergent component of motion, DCM) \(\xi=x+\dot x/\omega\) answers: "where must the foot land for the robot to come to rest exactly over it?" The answer is \(p=\xi\): since \(\dot\xi=\omega(\xi-p)\), a foot placed at \(\xi\) freezes \(\xi\), and \(x\) converges to \(\xi\) with time constant \(1/\omega\). But stepping takes time \(t_s\), during which \(\xi\) keeps growing as \(e^{\omega t_s}\), so the foot must actually land at \(\xi(t_s)=\xi_0e^{\omega t_s}\).

The feet can only reach so far (leg length, joint limits), which gives a capture region: for a mild push, shifting the ZMP to the toe with ankle torque is enough (the "ankle strategy", \(\xi_0\) within the foot); a larger push needs a step (the "stepping strategy"); a still larger one puts \(\xi(t_s)\) out of reach and the robot must fall. This is why a person who is pushed steps without thinking: the foot is small, and the ankle strategy only has a few centimetres of capacity.

取矢状面，\(x\) 向前、\(z\) 向上。整台机器人质量为 \(m\)，质心在 \((x,z)\)，地面反力的合力 \(\mathbf F=(F_x,F_z)\) 作用在地面上的一点 \(p\)（压力中心 CoP）。牛顿–欧拉方程给出 \(F_x=m\ddot x\)，\(F_z=m(\ddot z+g)\)，合力对质心的力矩等于角动量变化率 \(\dot H\)（逆时针为正）：

Work in the sagittal plane, \(x\) forward and \(z\) up. The whole robot has mass \(m\) and COM at \((x,z)\); the resultant ground reaction \(\mathbf F=(F_x,F_z)\) acts at one ground point \(p\) (the center of pressure, CoP). Newton–Euler gives \(F_x=m\ddot x\), \(F_z=m(\ddot z+g)\), and the moment of that resultant about the COM equals the rate of change of angular momentum \(\dot H\) (counter-clockwise positive):

这就是 ZMP（零力矩点）的表达式：它是地面上那一点，在它上面，重力与惯性力的合力矩的水平分量为零。LIPM 做了四个假设：(1) 质量集中在质心；(2) 质心高度恒定，\(z=z_c\)，\(\ddot z=0\)；(3) 腿无质量（摆动腿不影响质心）；(4) \(\dot H\approx0\)。代入即得

This is the expression for the ZMP (zero-moment point): the ground point about which the horizontal component of the moment of gravity plus inertial forces vanishes. The LIPM makes four assumptions: (1) all mass sits at the COM; (2) constant COM height, \(z=z_c\), \(\ddot z=0\); (3) massless legs (the swing leg does not affect the COM); (4) \(\dot H\approx0\). Substituting gives

闭式解可以直接代回方程验证，FIG 3 的读数把它与 RK4 数值积分比较（一步之内差在 \(10^{-10}\) m 以下）。\(E\) 叫轨道能量：\(\dot E=\dot x\ddot x-\omega^2(x-p)\dot x=0\)，所以在一步之内相平面轨迹是双曲线。\(E>0\) 说明质心会越过脚正上方继续前进，\(E<0\) 说明会在越过之前折返，\(E=0\) 恰好是两条流形。两条路得到同一个方程：对"刚性腿长 \(\ell\) 的摆"，\(x=\ell\sin\theta\)，\(\ddot\theta=\tfrac g\ell\sin\theta\)，小角度下 \(\sin\theta\approx\theta\)（2.4 节的线性化）给出 \(\omega=\sqrt{g/\ell}\)；LIPM 的"恒高度"假设则不要求角度小，但要求腿能伸缩来保持高度。

The closed form can be substituted back into the ODE to check, and FIG 3's readout compares it with an RK4 numerical integration (within one step they differ by less than \(10^{-10}\) m). \(E\) is the orbital energy: \(\dot E=\dot x\ddot x-\omega^2(x-p)\dot x=0\), so within one step the phase-plane path is a hyperbola. \(E>0\) means the COM will pass over the foot and keep going, \(E<0\) means it turns back before reaching it, and \(E=0\) is exactly the pair of manifolds. Two routes reach the same equation: for a rigid-leg pendulum of length \(\ell\), \(x=\ell\sin\theta\) and \(\ddot\theta=\tfrac g\ell\sin\theta\), and a small angle, \(\sin\theta\approx\theta\) (the linearization of Lesson 2.4), gives \(\omega=\sqrt{g/\ell}\); the "constant height" assumption of the LIPM does not need small angles but requires the leg to extend and retract to hold the height.

平地上，各接触点的法向力 \(f_{z,i}\ge0\)，压力中心是它们的加权平均：

On flat ground, with contact normal forces \(f_{z,i}\ge0\), the center of pressure is their weighted average:

凸组合必定落在接触点的凸包内，所以真实的 CoP 永远在支撑多边形里；平地上 ZMP 与 CoP 重合。FIG 2 的读数用 \(\sum x_if_{z,i}/\sum f_{z,i}\) 和上面的 ZMP 公式（其中 \(\dot H\) 取脚力对质心的合力矩）各算一遍：两者在代数上等价，差值只是浮点误差，它检验的是公式与代码一致，并不是一次独立的测量。如果某个运动"需要"的 ZMP 在多边形之外，那个点只是一个虚构的 ZMP：真实的 CoP 会停在边上，多出来的力矩变成绕这条边的角加速度，脚开始滚动，机器人开始翻倒。ZMP 条件只管法向力的单向性（\(f_z\ge0\)）；切向力的限制是另一个独立的条件，即 \(\|\mathbf f_t\|\le\mu f_z\)，另外还有绕竖直轴的摩擦力矩（偏航）。ZMP 在脚内，不保证不打滑。

A convex combination always lies in the convex hull of the contact points, so the real CoP can never leave the support polygon; on flat ground ZMP and CoP coincide. FIG 2's readout evaluates both \(\sum x_if_{z,i}/\sum f_{z,i}\) and the ZMP formula above (with \(\dot H\) taken as the moment of the foot forces about the COM): the two are algebraically equivalent, so their difference is round-off, a check that formula and code agree rather than an independent measurement. If a motion "needs" a ZMP outside the polygon, that point is only a fictitious ZMP: the real CoP sits on the edge, the leftover moment turns into angular acceleration about that edge, the foot starts to roll, and the robot starts to tip. The ZMP condition only encodes the one-sidedness of the normal force (\(f_z\ge0\)); limits on tangential force are a separate, independent condition, \(\|\mathbf f_t\|\le\mu f_z\), plus a friction limit on the moment about the vertical axis (yaw). A ZMP inside the foot does not guarantee that the foot does not slip.

令 \(\xi=x+\dot x/\omega\)，则 \(\dot\xi=\dot x+\ddot x/\omega=\dot x+\omega(x-p)=\omega\big(\hl{dcm}{\xi}-p\big)\)，同时 \(\dot x=-\omega(x-\xi)\)。所以 \(\xi\) 的方程是一个独立的一阶不稳定系统，而 \(x\) 只是它的一个低通滤波后的跟随者。单支撑相内

Let \(\xi=x+\dot x/\omega\). Then \(\dot\xi=\dot x+\ddot x/\omega=\dot x+\omega(x-p)=\omega\big(\hl{dcm}{\xi}-p\big)\), and also \(\dot x=-\omega(x-\xi)\). So \(\xi\) obeys its own first-order unstable ODE, and \(x\) is just a low-pass-filtered follower of \(\xi\). Within one single-support phase

周期步态：要求每一步开始时相对新脚的状态相同，且每步前进 \(L\)、历时 \(T\)。设开始时 \(x_0\)、\(\dot x_0\)（相对脚），结束时 \(x_T\)、\(\dot x_T\)。周期性要求 \(x_T-L=x_0\)，\(\dot x_T=\dot x_0\)，解得 \(x_0=-L/2\)，\(\dot x_0=\tfrac{\omega L}{2}\coth\tfrac{\omega T}{2}\)（FIG 3 的行走器正是从这组初值出发的，两个条件都成立到 \(10^{-16}\)）。平均速度 \(v=L/T\)；轨道能量 \(E=\tfrac12\omega^2(L/2)^2/\sinh^2(\omega T/2)\)。每步开始时相对新脚的捕获点是 \(b=\xi_{0}=\tfrac L2(\coth\tfrac{\omega T}{2}-1)\)，因此 DCM 落脚规则是

The periodic gait. Require every step to start from the same state relative to the new foot, with each step advancing \(L\) in time \(T\). Let the step start at \(x_0,\dot x_0\) (relative to the foot) and end at \(x_T,\dot x_T\). Periodicity demands \(x_T-L=x_0\) and \(\dot x_T=\dot x_0\), whose solution is \(x_0=-L/2\), \(\dot x_0=\tfrac{\omega L}{2}\coth\tfrac{\omega T}{2}\) (the simulator starts from exactly these values and both conditions hold to \(10^{-16}\)). The mean speed is \(v=L/T\) and the orbital energy \(E=\tfrac12\omega^2(L/2)^2/\sinh^2(\omega T/2)\). The capture point relative to the new foot at the start of each step is \(b=\xi_{0}=\tfrac L2(\coth\tfrac{\omega T}{2}-1)\), so the DCM foot-placement rule is

这一步把下一步开始时的 \(\xi\) 精确设成目标值（对不稳定模态是"一步到位"的），稳定模态每步以因子 \(e^{-\omega T}\approx0.2\) 衰减，所以 FIG 3 改变速度后两三步就进入新的周期步态。期望速度为 0 时 \(b=0\)，规则变成 \(p=\xi\)，正是捕获点。

This sets \(\xi\) at the start of the next step exactly to the target (a "dead-beat" move for the unstable mode), and the stable mode shrinks by \(e^{-\omega T}\approx0.2\) per step, which is why FIG 3 settles into a new periodic gait within two or three steps of a speed change. For zero desired speed \(b=0\) and the rule becomes \(p=\xi\), the capture point.

7.2 节：浮动基座机器人的构型是 \(q=({}^{W}T_{B},\boldsymbol\theta)\in SE(3)\times\mathbb R^n\)（速度 \(\dot q\) 取 \(6+n\) 维：基座 twist 与关节速度）。8.3 节的运动方程加上接触力，写成

Lesson 7.2: a floating-base robot's configuration is \(q=({}^{W}T_{B},\boldsymbol\theta)\in SE(3)\times\mathbb R^n\) (with generalized velocity of dimension \(6+n\): the base twist and the joint rates). The equations of motion of Lesson 8.3 plus contact forces read

\(S^\top\boldsymbol\tau\) 的前 6 个分量恒为 0：电机只作用在关节上。所以前 6 行（基座方向）是 \(M_{bb}\dot{\mathbf v}_b+M_{b\theta}\ddot{\boldsymbol\theta}+h_b=\sum_iJ_{c,i,b}^{\top}\mathbf f_i\)：基座加速度只能通过摆动关节带来的惯性耦合和接触力改变。接触力本身又由 \(J_c\dot{\mathbf v}+\dot J_c\mathbf v=0\)（脚不滑动）这类约束和关节力矩共同决定（8.4 节）。把前 6 行写成整体的力与力矩（所有关节的内力抵消），就得到 \(\sum_i\mathbf f_i+m\mathbf g=m\ddot{\mathbf c}\) 与 \(\dot{\mathbf h}_G=\sum_i(\mathbf p_i-\mathbf c)\times\mathbf f_i\)（加上脚上的接触力矩）：这是质心动力学（centroidal dynamics）。地面力能产生的净 wrench（7.6 节）构成一个凸锥 \(\{\sum_iG_i\mathbf f_i:\mathbf f_i\in\mathcal F_i\}\)，足式 MPC 和全身控制就是在这个锥里选 \(\mathbf f_i\)（Part IV，可写成 QP，6.8 节）。

The first six components of \(S^\top\boldsymbol\tau\) are identically zero: motors only act on joints. The first six rows (the base directions) read \(M_{bb}\dot{\mathbf v}_b+M_{b\theta}\ddot{\boldsymbol\theta}+h_b=\sum_iJ_{c,i,b}^{\top}\mathbf f_i\): the base acceleration can change only through inertial coupling from moving the joints and through contact forces. The contact forces themselves are determined jointly by constraints such as \(J_c\dot{\mathbf v}+\dot J_c\mathbf v=0\) (no foot slip) and the joint torques (Lesson 8.4). Summing the first six rows as a whole-body force and moment (all internal joint forces cancel) gives \(\sum_i\mathbf f_i+m\mathbf g=m\ddot{\mathbf c}\) and \(\dot{\mathbf h}_G=\sum_i(\mathbf p_i-\mathbf c)\times\mathbf f_i\) (plus contact torques at the feet): the centroidal dynamics. The net wrench (Lesson 7.6) the ground can produce forms a convex cone \(\{\sum_iG_i\mathbf f_i:\mathbf f_i\in\mathcal F_i\}\), and legged MPC and whole-body control choose the \(\mathbf f_i\) inside it (Part IV, expressible as QPs, Lesson 6.8).

Raibert（1986）的单腿跳跃机用一条朴素的规则平衡速度：脚落在"中性点" \(\dot x\,T_s/2\) 前方，再加一个速度反馈项，\(x_{\text{foot}}=\dot x\,\tfrac{T_s}{2}+k(\dot x-\dot x_{\text{des}})\)，\(T_s\) 是支撑时间。形式上它和 DCM 落脚规则是近亲。把腿换成弹簧，就得到跑步的模型 SLIP（弹簧加载倒立摆）：质心在支撑相像在弹簧上弹跳，腾空相是抛体。LIPM 的支撑相不做功（\(E\) 不变）；真实机器人落脚时有碰撞损失、蹬地时补充能量，被动行走机（McGeer 1990）则靠斜坡上的重力势能补回。

Raibert's (1986) one-legged hopper balanced its speed with a plain rule: place the foot ahead of the "neutral point" \(\dot x\,T_s/2\), plus a speed-feedback term, \(x_{\text{foot}}=\dot x\,\tfrac{T_s}{2}+k(\dot x-\dot x_{\text{des}})\), with \(T_s\) the stance time. It is a close cousin of the DCM placement rule. Replace the leg by a spring and you get the running model SLIP (spring-loaded inverted pendulum): the COM bounces on the spring during stance and flies ballistically during flight. LIPM stance does no work (\(E\) is constant); real robots lose energy at foot impact and put it back by pushing off, and passive-dynamic walkers (McGeer 1990) get it back from gravity on a slope.

四足动物（和四足机器人）按固定的节奏轮流抬腿，这个节奏叫步态。每条腿在一个周期里有一段支撑相（脚在地上，占周期的比例叫占空比 \(\beta\)）和一段摆动相，四条腿的周期之间错开一定的相位。下图左上是步态图（Hildebrand 图）：每行一条腿，实心是支撑，空白是摆动；左下画出每个时刻支撑多边形能给出的最大静态稳定裕度（多边形内最大内切圆的半径）；右侧是俯视动画：实心圆是支撑脚，空心圆是摆动脚，紫环是理想的静态质心位置（质心在身体里，所以机器人必须在迈步之前先晃动身体，把质心挪过去）。

Quadrupeds (animal or robot) lift their legs in a fixed rhythm called a gait. In each cycle every leg has a stance phase (foot on the ground; its fraction of the cycle is the duty factor \(\beta\)) and a swing phase, and the four legs' cycles are offset by phases. Top left is the gait diagram (a Hildebrand diagram): one row per leg, filled for stance and blank for swing. Below it is the largest static stability margin the support polygon can offer at each instant (the radius of its largest inscribed circle). On the right is a top-view animation: filled circles are stance feet, hollow circles swing feet, and the purple ring is the ideal static COM position (the COM lives in the body, so the robot must sway its body to bring the COM there before stepping).