Robot control
Geometric control
Definition
Geometric control designs control laws using the intrinsic geometry of a system's configuration space rather than treating its state as unconstrained coordinates. In robotics, it is often used for rotation and rigid-body motion on manifolds such as SO(3) and SE(3).
Also known as: Geometric nonlinear control
Updated
Design on the configuration space
A robot's orientation is not an arbitrary three-dimensional vector. Rotations belong to the special orthogonal group SO(3), and a rigid pose belongs to the special Euclidean group SE(3). A geometric controller formulates errors, dynamics, and feedback directly on such spaces so that the commanded state remains a valid rotation or pose.
The quadrotor control paper by Lee, Leok, and McClamroch models rigid-body dynamics on SE(3) and constructs nonlinear tracking controllers for attitude, position, and velocity flight modes. Its analysed results are almost global rather than global, which reflects the topology of three-dimensional attitude rather than merely a tuning choice.
Geometry is more than a pose representation
A rotation matrix or quaternion can represent orientation inside many kinds of controller. Geometric control refers to how the control law and its analysis respect the structure of the configuration space. It is not simply the use of one representation.
This approach can avoid coordinate singularities associated with local angle charts and can express errors without switching between local parameterisations. It is used for aerial vehicles, spacecraft, mobile robots, and manipulators whose motions include coupled translation and rotation.
Topology and the physical model remain limiting
Working on a manifold does not make every initial condition globally stabilisable by a smooth time-invariant feedback law. Controllers may provide almost-global convergence and retain exceptional initial states. Implementations also need numerically consistent integration and re-normalisation so finite-precision calculations do not drift away from the represented group.
Geometric design does not remove model mismatch, input saturation, contact uncertainty, or state-estimation error. A mathematically valid desired pose may still be unreachable or collide with the environment. The controller must be combined with feasible trajectory optimization, actuator limits, and task-specific safety handling.
Sources
Related terms
Rotation matrix
A rotation matrix represents an orientation or rotation while preserving lengths and angles. In three dimensions it is a 3-by-3 orthonormal matrix with determinant positive one.
Quaternion
A quaternion is a four-component mathematical object consisting of a scalar and a three-component vector. Robotics commonly uses unit quaternions to represent three-dimensional rotations without the coordinate singularities of Euler angles.
Rigid-body transformation
A rigid-body transformation changes a body's position and orientation without changing its shape or size. In three-dimensional robotics it consists of a proper rotation and a translation.
Trajectory optimization
Trajectory optimization finds a time-varying motion, and often control inputs, that minimizes an objective while satisfying specified constraints. Robot applications can include geometric, kinematic, and dynamic constraints.