cs.ROOct 12, 2026

Design and Analysis of a Nonlinear Parallel Mechanism Coupling for Dynamic Humanoid Legs

Authors: Rohit Kumar, Heiner Peters, Felix Wiebe, Franek Stark, Janne Janzen, Dennis Mronga, Shubham Vyas, Frank Kirchner

Organizations: Robotics Innovation Center, DFKI GmbH, 28359 Bremen, Germany. · AG Robotik, University of Bremen, 28359 Bremen, Germany.

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Abstract

Humanoids increasingly demand lightweight leg mechanisms for agile, dynamic, and energy-efficient locomotion. Reducing distal inertia through proximal actuation has thus become a common design goal. Yet, compact transmissions that preserve torque capability and efficient force transmission remain hard to achieve. This paper presents a novel series-parallel leg design combining a knee parallel mechanism, an ankle parallel mechanism, and a nonlinear knee-ankle coupling where all actuators placed closer to the hip. As the knee flexes, the coupling passively assists ankle dorsiflexion for foot clearance and, as it extends it assists plantarflexion for push-off, redistributing ankle load through the knee transmission. We derive the coupling's nonlinear transmission models and validate them in simulation and on hardware. The design shows that the shared transmission lets both joints draw on each other's actuator capacity by easing the knee's peak torque demand under load and the ankle's peak speed demand during fast motion. With the leg's proximal actuator placement, the leg achieves an effective reflected mass of 0.37 kg at the foot for a 6 kg leg, demonstrating very low distal leg inertia for dynamic humanoid robots.

Figures & tables

Figure 1: CAD view (left), parallel mechanisms (middle), and real system (right). Knee mechanism is shown in yellow and ankle mechanism is shown in blue with belts shown in red.
TypePositionVelocityAcceleration
Implicitϕ(q)=0\bm{\phi}(\mathbf{q})=\mathbf{0}Kq˙=0\mathbf{K}\dot{\mathbf{q}}=\mathbf{0}Kq¨=k\mathbf{K}\ddot{\mathbf{q}}=\mathbf{k}
Explicitq=γ(y)\mathbf{q}=\bm{\gamma}(\mathbf{y})q˙=Gy˙\dot{\mathbf{q}}=\mathbf{G}\dot{\mathbf{y}}q¨=Gy¨+g\ddot{\mathbf{q}}=\mathbf{G}\ddot{\mathbf{y}}+\mathbf{g}
Table I: Loop closure constraints [ 13 ]
Figure 2: Important dimensions for the leg (blue denotes the center of mass location), with the mass distribution of the leg.
Figure 3: Finite element analysis with 600 N vertical force at the foot.
Figure 4: Parallel mechanism (left) and topological graph (right) of the leg.
Figure 5: Knee submechanism with joint notations (left) and configuration space (right).
Figure 6: Maximum knee velocity (left) and torque (right) for different configurations.
Figure 7: Ankle joints notation (left) and feasible configuration space (right).
Figure 8: Maximum ankle pitch velocity (left) and roll velocity (right) across the ankle configuration space.
Figure 9: Maximum ankle pitch torque (left) and roll torque (right) across the ankle configuration space.
Figure 10: Coupling joints notation (left) and configuration space of the ankle pitch with knee (right).
Figure 11: Velocity-level coupling between the knee and ankle joints.
Figure 12: Independent joint torques (left) and motor torques (right) vs foot vertical position during the squat.
Figure 13: Coupling verification with an external load of 2.5kg at 0.5m distance.
Figure 14: Velocity tracking for sinusoidal trajectory of the foot.
Figure 15: Mean reflected mass distribution for randomly sampled configurations in the legs’ configuration spaces.
RobotLlegL_{\mathrm{leg}} [m]mlegm_{\mathrm{leg}} [kg]Ihip/(mlegLleg2)I_{\mathrm{hip}}/(m_{\mathrm{leg}}L_{\mathrm{leg}}^{2})mˉn^/mleg\bar{m}_{\hat{n}}/m_{\mathrm{leg}}
Our design0.7226.070.054 ±\pm 0.0170.061 ±\pm 0.008
MIT Humanoid0.7115.330.053 ±\pm 0.0150.145 ±\pm 0.035
Kangaroo1.03413.180.057 ±\pm 0.0170.075 ±\pm 0.005
Table II: Leg length, mass, normalized swing inertia about the hip-pitch axis, and normalized MRM at the foot
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