Whole-Body Operational Space Control

• WBOSC is a robotic control paradigm that integrates task-space motion with internal force regulation, ensuring precise operations in underactuated, contact-rich environments.
• It splits control efforts between motion generation and internal force management, using sensor-based feedback and series elastic actuators for accuracy.
• Experimental validations show WBOSC enables agile, dynamically stable locomotion by continually updating trajectories and correcting force deviations in complex terrains.

Whole-Body Operational Space Control (WBOSC) is an advanced robotic control paradigm that synthesizes task-space control, force regulation at contacts, and consistent handling of redundancy for floating-base, contact-interacting robots. WBOSC extends classic operational space control to high-DoF, underactuated systems that require simultaneous execution of motion and force tasks under physical constraints. Its structure separates motion generation in task space from the control of internal forces that regulate multi-contact dynamics, making it pivotal for agile, dynamically stable behaviors in humanoid and legged robots.

1. Theoretical Foundation and Control Structure

WBOSC generalizes the operational space control framework for floating-base robots experiencing multiple contacts by splitting the control torques into orthogonal subspaces: one responsible for producing the desired operational (task-space) acceleration and the other for regulating nullspace “internal forces.” The fundamental torque command structure is: $\tau_{control} = J^{*T} F + L^{*T} \tau_{int}$ where:

• $J^*$ is the contact-consistent task Jacobian mapping joint velocities to task-space variables (e.g., COM position, swing foot).
• $F$ is the task-space force command generated by a model-based feedback law, including compensation for inertia, Coriolis, and gravity effects.
• $L^*$ is the null-space projection onto internal forces, defined such that $L^{*T} \tau_{int}$ does not induce net robot motion but modulates internal load sharing between contacts.
• $\tau_{int}$ is the internal force command.

The task control law for a desired task-space trajectory $p^d(t)$ uses: $F = \Lambda^* u + \mu^* + p^*$ where $\Lambda^*$ is the operational-space inertia, $\mu^*$ the Coriolis/centrifugal terms, and $p^*$ the gravity compensation, with $u$ chosen via PD+integral control to minimize tracking error. The strict orthogonality between task-space control and internal force regulation is essential for stable and efficient whole-body behavior, especially in multi-contact scenarios(Kim et al., 2015).

2. Internal Force Regulation and Sensor-Based Feedback

A defining advance of WBOSC is the explicit control of internal forces, meaning those force components at contacts that, by construction, produce no net motion or momentum but regulate interaction between contact points (e.g., tension between biped feet). This is particularly crucial for tasks such as balancing on split or inclined terrain, where sufficient internal force must be maintained to prevent slipping.

The internal force is computed as: $F_{int} = W_{int} F_r$ where $F_r$ is the vector of contact reaction forces and $W_{int}$ is a mapping (selection, rotation, differencing) which extracts the relevant internal force component. On a biped, for example, this involves projecting foot forces onto the line joining the feet and differencing the results.

The internal force control command is: $$\Gamma_{int} = J^{*T}_{i|l} [ F_{int,ref} - F_{int,\{t\}} + \mu_i + p_i + K_F (F_{int,ref} - F_{int,act}) ]$$ where $F_{int,ref}$ is the desired internal force, $F_{int,act}$ is the measured internal force (computed from high-fidelity actuator torques, typically via series elastic actuators), and $K_F$ is a feedback gain. This sensor-based feedback regulation is critical to compensate for modeling inaccuracies and to robustly maintain internal force levels as high as 100 N during aggressive disturbances or on steep split terrain(Kim et al., 2015).

3. Role of Series Elastic Actuators (SEAs)

Series elastic actuators are integral to WBOSC implementations where high-fidelity force/torque sensing is required. The embedded compliance provides two key benefits:

• The deformation of the elastic element offers a direct and accurate measurement of the actuator output torque.
• The reduced output inertia of SEAs improves force tracking during dynamic contact transitions, such as swing leg landings.

Challenges arise due to actuator compliance—which may reduce the efficiency of swing leg trajectories—but are addressed via gain scheduling, assigning higher torque gains to swing legs and lower gains to stance legs to maintain both force precision and trajectory responsiveness(Kim et al., 2015).

4. Trajectory Generation and Online Footstep Planning

For underactuated robots (such as point-foot bipeds), the control strategy is coupled with an online trajectory generator based on the prismatic inverted pendulum model (PIPM):

$$\frac{d^2x}{dt^2} = \frac{g + \frac{d^2z}{dx^2} \left( \frac{dx}{dt} \right)^2}{z - (x - x_p) \frac{dz}{dx}}$$

where $x_p$ denotes stance foot position and $z=h(x)$ the COM height surface.

Key steps:

1. Observe the real-time COM state (position, velocity) as the robot steps.
2. Forward integrate PIPM to the predicted foot switch event, including an empirical velocity drop at impact.
3. Given a constant "time to velocity reversal" $t'$, solve (typically via bisection) for the next foot placement such that the COM velocity reverses direction in $t'$.
4. Repeat this discrete-time re-planning at every step to stabilize the inherently unstable point-foot biped locomotion via a closed feedback loop.

This method supports robust undirected walking, where the robot is not given a fixed path but is stabilized through actively re-planned footsteps, allowing real-time corrections for model errors or disturbances(Kim et al., 2015).

5. Challenges in Multi-Contact and Underactuated Scenarios

WBOSC directly addresses key challenges encountered in difficult environments:

• On split terrain with high pitch (e.g., dual 45° wedges), precise regulation of internal forces maintains sufficient friction for stability. If the internal force drops (e.g., toward 10 N), the robot is prone to slip and fall. Maintaining ≈100 N ensures robust frictional contact.
• The separation of motion and force control allows the system to rapidly correct for force deviations or external disturbances, such as pushes.
• In point-foot bipedal walking, underactuation makes direct COM control impossible during single support. The continuous update of footstep plans, integrated with WBOSC's accurate continuous feedback, overcomes this limitation.
• Transitions between stance and swing are smoothed by state-dependent ramping of reaction force targets, mitigating torque discontinuities that could destabilize the robot(Kim et al., 2015).

6. Experimental Validation and Performance

The WBOSC approach, as implemented, was rigorously tested and analyzed:

• On a point-foot bipedal robot with series elastic actuators, WBOSC enabled balancing on split terrain with precise internal force regulation. Maintaining target internal force (≈100 N) prevented slip and allowed rapid correction for external disturbances.
• In stepping and walking experiments, the combination of WBOSC control and online trajectory generation (via a phase-space planner) stabilized the robot under discrete-time foot placement updates, successfully achieving robust undirected walking.
• During dynamic locomotion with a planarized setup, deviations (e.g., 5.5 cm in one footstep) were corrected in subsequent steps, demonstrating the closed feedback nature of the architecture, both at the trajectory planning and operational space layers.

These results concretely demonstrate the WBOSC paradigm's effectiveness for achieving and maintaining agile, dynamically stable locomotion in complex environments(Kim et al., 2015).

7. Summary and Key Equations

A summary of the core equations underpinning WBOSC in this context is as follows:

Component	Mathematical Expression	Description
Whole-body control torque	$\tau_{control} = J^{*T}F + L^{*T} \tau_{int}$	Task-space motion + internal force regulation
Operational space force law	$F = \Lambda^*u + \mu^* + p^*$	PD control plus model-based compensation
Internal force correction	$$\Gamma_{int} = J^{*T}_{i|l}[F_{int,ref} - F_{int,\{t\}} + \mu_i + p_i + K_F (F_{int,ref} - F_{int,act})]$$	Sensor-based internal force feedback
PIPM COM dynamics	$$d^2 x / dt^2 = \frac{g + (d^2 z/dx^2) (dx/dt)^2}{z - (x - x_p)(dz/dx)}$$	Used in online trajectory generation

The WBOSC methodology, integrating these approaches, enables robots to handle the delicate balance between agile motion and precise force control in complex multi-contact and underactuated scenarios, providing a versatile control foundation for advanced humanoid locomotion and manipulation(Kim et al., 2015).