Humanoid Robot: Balance, Joint Torque and Runtime

The model combines one single-support balance step, sagittal stance load moments and an independent electrical energy budget.

Kajita et al. (2003), attached source s1, page 2, equations (6)–(10), gives $p_x=x-h\ddot{x}/g$ and $p_y=y-h\ddot{y}/g$ for constant CoM height and negligible centroidal angular-momentum rate. Here $\omega=\sqrt{g/h}$ and $T=S/v$. The symmetric nominal trajectory is

With initial position offsets and acceleration biases $b_x,b_y$,

Thus the required ZMP is constant: $p_x=x_0-b_x/\omega^2$, $p_y=y_0-b_y/\omega^2$. The nominal initial forward velocity is $S\omega/[2\tanh(\omega T/2)]$; lateral initial velocity is zero. Only the unbiased step has mean forward speed equal to the nominal mission speed.

Accelerations are evaluated as analytic second derivatives: $\ddot{x}=\ddot{x}_n+b_x\cosh(\omega t)$ and $\ddot{y}=b_y\cosh(\omega t)$. An independent four-state ode45 solution integrates $\dot{x}=v_x$, $\dot{v}_x=g(x-p_x)/h$ and the corresponding lateral equations from the prescribed initial state. Property checks compare the full trajectories, accelerations, and initial/final velocities, sweeping nonzero forward and lateral biases and nominal speed. The agreement tolerances are 2 μm for position, 5×10⁻⁵ m/s² for acceleration and 2 μm/s for endpoint velocity.

The sole margin is $\min(L/2-|p_x|,W/2-|p_y|)$. The model requires $F_z=mg$ and $F_x=m\ddot{x}$. Available friction is compared with $\max\sqrt{\ddot{x}^2+\ddot{y}^2}/g$.

A forward-bending knee is obtained by two-circle intersection using the thigh and shank lengths. At each joint $j$, the sagittal ground-reaction load moment is

The balancing actuator torque has the opposite sign. Peak cards use magnitudes across 201 evenly spaced stance samples, multiplied by the user sizing allowance. Swing-knee floor clearance is checked over all 201 step samples and at the current animated pose; a negative clearance blocks results. No full multibody inverse dynamics is claimed.

Collins et al. (2005), s2 page 2, defines dimensionless electrical CoT as energy divided by weight and distance, and distinguishes it from positive mechanical work. Here $e_m=C_w mg$, $P_m=e_m v$, $P_{walk}=P_m+P_{aux}$, and $P_{mission}=dP_m+P_{aux}$. Available energy is $E_u=E_{pack}f_u$; runtime is $E_u/P$ and mission walking distance is $vdt$. Use explicit kWh/Wh and hour/second conversions. The CoT input is incremental: a total published electrical CoT requires $P_{aux}=0$.

• Generic, unofficial robot inputs; no manufacturer presets or inferred specifications.
• Flat, rigid rectangular stance sole, ankle centered in it, no slip and no vertical CoM acceleration. Single support carries the full combined weight.
• Robot plus payload is represented by one mass at constant height; the hip is directly below it. Both leg links are massless. Lateral moments and link inertias are omitted.
• The illustrated swing ankle follows a smooth clearance path proportional to leg length. Its inverse-kinematic pose is schematic and contributes no torque or energy. Footprint lateral spacing is schematic.
• Replaying the animation resets a single step. Impact, double support, push-off, support exchange and whole-gait feedback are not simulated. The lateral initial condition is not a periodic gait controller.
• Electrical CoT stays constant for the mission and the speed sweep. Joint geometry and electrical CoT are separate estimates; posture changes require a measured or otherwise justified CoT update.
• Always-on power persists during stationary time and walking. Reserve, pack aging and other energy derating are combined in the usable fraction. No extra efficiency is applied to electrical CoT.

Use for preliminary comparisons and teaching, on level ground with moderate lateral motion. ZMP inside the sole is a necessary contact-feasibility condition under these assumptions; it is not proof of stability, capturability or a feasible whole-body gait. Tedrake's Underactuated Robotics, chapter 5, explains these limits.

Correct all reach and knee-clearance errors before interpreting joint loads. Swing-knee floor clearance is checked at all 201 step samples and at the current animated pose. A longer support interval amplifies bias exponentially; the tool warns when $\omega T>3$. The sagittal load estimate is flagged for lateral excursions above 20% of CoM height. Friction is a separate necessary contact check.

The sampled stance moments exclude link gravity/inertia, swing loads, impact peaks, compliance, ankle sizing, torso angular momentum, joint speed and motor/gearbox limits. A sizing multiplier cannot replace full inverse dynamics, structural analysis or actuator thermal evaluation. The results do not establish a safe robot design.

The energy budget excludes manipulation, slopes, starts/stops and charging. Battery sag, discharge-rate limits, cold-temperature performance and changing compute loads require measured corrections. At zero walking fraction and zero always-on power, a finite runtime is undefined and the tool asks for a positive load.

The Cornell and ASIMO checks compare published electrical estimates only; their generic leg geometry is not a reconstruction of those robots. The Cornell paper's rounded CoT predicts 10.20 W versus its measured 11 W. For ASIMO, its approximate CoT 3.2 gives 0.529 h versus about 0.5 h; the reported raw battery/speed numbers imply CoT about 3.39. These differences are documented in the reference cases rather than removed by changing physics.

Industry context is dated: TrendForce, 28 January 2026; 1X NEO manufacturer page checked 27 October 2026. A vendor's quoted runtime lacks the duty-cycle definition needed to validate a continuous walking prediction.