Underactuation

An underactuated system is defined by the inequality where the number of independently controllable actuators is strictly less than the system's degrees of freedom (DoF). For example, a simple 2-link planar arm with 2 rotational joints (2 DoF) but only one motor is underactuated. In legged locomotion, a robot's floating base (6 DoF: 3 position, 3 orientation) is never directly actuated, making all legged robots fundamentally underactuated during flight phases. Control must therefore rely on indirect forces from contact with the environment.

Control cannot be achieved through direct joint-by-joint positioning. Instead, controllers must exploit dynamic coupling and non-holonomic constraints to induce motion in unactuated directions. For instance, a cart-pole system (inverted pendulum) uses horizontal cart motion to control the unactuated pole angle. In legged robots, swinging one leg creates reaction forces that can be used to control torso orientation. This often requires solving nonlinear, underactuated optimal control problems to find feasible trajectories.

Underactuated designs frequently incorporate passive elements like springs, dampers, or compliant joints to store and release energy, aiding control. The Spring-Loaded Inverted Pendulum (SLIP) model is a classic example, where a passive spring leg enables stable running. Series Elastic Actuators (SEA) introduce intentional compliance. In passive dynamic walkers, gravity and natural pendulum dynamics produce stable walking down a slope with no motors, showcasing the ultimate reliance on passive dynamics.

Achieving full control often depends on intermittent contact with the environment. The forces generated at contact points (e.g., feet touching the ground) provide the necessary constraints and reaction forces to influence unactuated states. This makes contact timing, foot placement, and ground reaction force (GRF) control paramount. Strategies like Capture Point and Divergent Component of Motion (DCM) explicitly plan foot placements to manage the underactuated center of mass dynamics.

Standard linear control techniques often fail. Success typically requires nonlinear control methods like partial feedback linearization, energy shaping, or model predictive control (MPC). Engineers use reduced-order models (ROMs) like the Linear Inverted Pendulum (LIPM) to capture the essential underactuated dynamics for planning, ignoring many actuated degrees of freedom. The control hierarchy then uses whole-body control (WBC) to map these simple model commands to the full robot's actuators.

While challenging to control, underactuation offers significant advantages:

• Energy Efficiency: Exploiting natural dynamics and passive elements reduces actuator effort (e.g., passive dynamic walkers have extremely low Cost of Transport).
• Mechanical Robustness: Fewer actuators mean lower weight, cost, and potential points of failure.
• Natural, Compliant Motion: Systems can exhibit smooth, animal-like movement and are often more compliant to external disturbances, improving safety in human interaction.
• Agility: Dynamic coupling allows for rapid, whole-body maneuvers like throwing the torso to take a step.

These are the quintessential example of underactuation. A passive dynamic walker is a purely mechanical, unpowered device that can walk stably down a shallow slope. It has no motors or computers; its gait emerges entirely from the interaction of its dynamics, gravity, and the ground. Key features include:

• Energy efficiency: Converts potential energy directly into motion.
• Natural gait: Exhibits human-like, heel-to-toe rolling motion.
• Minimalist design: Proves stable locomotion does not require complex, fully actuated control for each joint.

Humanoid robots like Boston Dynamics' Atlas often use point feet (or very small foot areas). This design choice is intentionally underactuated at the ankle. The robot cannot directly control the pitch and roll of its foot relative to the ground. Instead, it must use its whole-body dynamics and momentum to maintain balance. Control is achieved through:

• Dynamic balancing: Using angular momentum from the torso and arms.
• Precise foot placement: Planning steps to keep the Center of Pressure within the tiny support area.
• Compliant ankles: Often using Series Elastic Actuators to absorb shocks and provide ground feel.

During a trot—a diagonal gait where two legs are in the air simultaneously—a quadruped robot becomes underactuated. With only two legs in contact, it cannot independently control all six degrees of freedom of its torso (position and orientation). The system is dynamically coupled. Stability is maintained by:

• The Spring-Loaded Inverted Pendulum (SLIP) model: Treating the body as a bouncing mass on a virtual spring.
• Orbital energy regulation: Controlling the total energy of the system to maintain a periodic hopping motion.
• Rapid leg adjustment: Using the swing legs to apply corrective forces upon landing.

Underwater gliders are a marine robotics example. They typically have only one or two actuators (a pump to change buoyancy and a movable mass to adjust pitch). They cannot directly propel themselves forward. Instead, they exploit hydrodynamic forces:

• Buoyancy engine: The pump changes volume, making the glider sink or rise.
• Wings: Convert vertical motion into forward glide via lift.
• Passive stability: The hull and fin design provide inherent directional stability. This extreme underactuation allows for missions lasting months, covering thousands of kilometers on minimal battery power.

While not a complete robot, Series Elastic Actuation is a component-level design that introduces intentional underactuation. A spring is placed between the motor and the output link. The motor controls the spring deflection, not the output position directly. This creates a compliant, underactuated joint that provides critical benefits:

• Force control & safety: The spring acts as a force sensor and a mechanical buffer.
• Energy storage: Enables efficient dynamic motions like running by recycling energy.
• Shock absorption: Protects gears from impact loads. This principle is central to robots like MIT's Cheetah.

Many agile mobile robots, like the MIT Mini Cheetah in certain modes, use dynamical steering instead of a steered axle. The robot turns by applying differential torques to the left and right legs/ wheels and using its body's yaw inertia and centripetal forces. This is underactuated because there is no dedicated steering actuator; yaw control is achieved through the dynamic coupling of lateral and forward motion. Advantages include:

• Mechanical simplicity: Fewer parts and lower weight.
• High-speed agility: Enables rapid, drift-like turns.
• Fault tolerance: Can still turn if a dedicated steering actuator fails.

A mode of locomotion where a legged system walks down a shallow slope using only gravity and its natural dynamics, with minimal or no actuation. This is the quintessential example of underactuation in motion, demonstrating how mechanical design can exploit dynamics to produce stable, energy-efficient gaits without full control over every joint.

• Key Principle: Relies on the natural pendulum-like dynamics of the legs and torso.
• Connection to Underactuation: These walkers are often highly underactuated or completely passive, proving that stable locomotion does not require direct actuation of every degree of freedom.

A simplified dynamic representation, such as the Linear Inverted Pendulum (LIP) or Spring-Loaded Inverted Pendulum (SLIP), that captures the essential dynamics of a complex legged robot for planning and control.

• Purpose: Enables tractable computation for high-level control by ignoring the full kinematics of every limb.
• Connection to Underactuation: Control is often designed in the low-dimensional space of the ROM. The controller must then map these simplified commands back to the full, underactuated system, a process that inherently deals with the lack of direct control over all degrees of freedom.

A state variable derived from the Linear Inverted Pendulum Model that captures the unstable part of the center of mass dynamics. It is used for planning stable foot placements for bipedal robots.

• Function: The DCM naturally diverges from the center of mass; placing a footstep at its projected location can stabilize the system.
• Connection to Underactuation: In an underactuated biped, you cannot instantaneously correct the center of mass velocity with torque at the ankle. The DCM framework provides a method to plan the one control action you do have—foot placement—to manage the unstable dynamics inherent in underactuated balance.

The dynamics describing the relationship between the net external wrenches (forces and moments) acting on a robot and the motion of its center of mass and its centroidal angular momentum.

• Critical for Underactuation: For a legged robot with a floating base, the six degrees of freedom of the base (position and orientation) are unactuated. Their motion is governed entirely by the ground reaction forces at the feet.
• Control Implication: In underactuated systems, you control these base dynamics indirectly by manipulating contact forces. Whole-body controllers use centroidal dynamics as a high-priority task to ensure dynamic feasibility.

A hierarchical control framework that coordinates all of a robot's degrees of freedom to execute multiple tasks (e.g., balance, foot tracking) while respecting physical constraints like torque limits and contact forces.

• How it Manages Underactuation: WBC typically formulates control as a Quadratic Program (QP). The unactuated floating base dynamics are treated as hard constraints that must be satisfied. The optimizer then solves for joint torques and contact forces that achieve desired tasks as closely as possible within these fundamental dynamic constraints.
• Result: It provides a systematic method to generate control commands for a fully-actuated kinematic chain that is dynamically underactuated at its base.

The equations of motion for a multi-body system, like a legged robot, where the base link (e.g., the torso) is not fixed to the world and has six unactuated degrees of freedom (three translational, three rotational).

• The Source of Underactuation: This floating base is the primary reason a legged robot is considered underactuated. You cannot apply a direct torque to translate the base or rotate it in mid-air.
• Mathematical Form: The dynamics are split into actuated (joint) and unactuated (base) parts: M(q)v̇ + h(q, v) = Sᵀτ + Jᶜᵀ(q)λ. The matrix S selects the actuated dimensions, highlighting the lack of direct control over the base acceleration.