Spatial Vector Algebra

A spatial velocity is a 6D vector that combines the linear velocity of a point on a rigid body (typically its center of mass) and its angular velocity into a single entity. Formally, it is represented as v = [ω; v_lin], where ω is the 3D angular velocity and v_lin is the 3D linear velocity. This unified representation allows for concise expression of kinematic relationships across a kinematic chain, such as computing the velocity of a robot's end-effector from its joint velocities using a spatial Jacobian.

A spatial force (or wrench) is a 6D vector that combines a torque and a force applied to a rigid body. It is the dual of spatial velocity. Formally, it is f = [τ; f_lin], where τ is the 3D torque and f_lin is the 3D force. This representation simplifies the expression of dynamics, such as in the Newton-Euler equations, where the net spatial force equals the derivative of the spatial momentum. It is essential for calculating joint torques in inverse dynamics.

A spatial transform is a 6x6 matrix that maps spatial vectors (velocities, forces, accelerations) from one coordinate frame to another. It is constructed from the 3D rotation matrix E and the translation vector r between frames. For velocities, the transform is X = [[E, 0]; [ -E * [r]×, E ]], where [r]× is the skew-symmetric cross-product matrix. This transform is crucial for propagating motion and force quantities across the links of a robot or articulated mechanism.

The spatial inertia matrix is a 6x6 symmetric matrix that encapsulates the mass distribution of a rigid body. It combines the body's mass, center of mass location, and its inertia tensor about a point. Formally, for a body with mass m, center of mass c, and inertia tensor I_c about the center of mass, the spatial inertia I about a point O is I = [[ I_c + m * [c]× * [c]×ᵀ, m * [c]× ]; [ m * [c]×ᵀ, m * I₃ ]]. This matrix directly relates spatial acceleration to spatial force via f = I * a.

Spatial vector algebra defines two specialized cross-product operators that simplify equations of motion:

• Spatial Cross Product for Motion Vectors (v × m): Used in velocity transformations and the time derivative of spatial transforms.
• Spatial Cross Product for Force Vectors (v ×* f): The dual of the motion cross product, used in the Newton-Euler equations to account for gyroscopic and Coriolis forces. These operators are represented by 6x6 matrices built from the components of the spatial velocity v. They are key to writing compact, coordinate-free dynamic equations.

The Featherstone algorithm (Articulated Body Algorithm) uses spatial vectors to compute forward dynamics for complex robotic chains in O(n) time. This is the computational backbone for simulating humanoid robots, robotic arms, and walking machines. It recursively propagates forces and accelerations through the kinematic tree using spatial transforms, avoiding the costly inversion of a large system mass matrix.

• Key Concept: Uses spatial inertia and articulated body inertia.
• Example: Simulating a 7-degree-of-freedom robotic arm for motion planning.

Spatial vectors elegantly model contact forces and friction impulses in multi-body systems. The 6D spatial force vector combines linear force and torque, allowing a single equation to represent the wrench at a contact point. This is critical for physics engines simulating pushing, grasping, or walking on uneven terrain.

• Key Concept: A contact impulse is applied as a spatial vector.
• Integration: Used within Linear Complementarity Problem (LCP) solvers to compute non-penetrating contact forces.

This robotics control paradigm uses spatial algebra to compute joint torques that produce desired end-effector spatial accelerations. It relies on the dynamically consistent generalized inverse of the Jacobian matrix, which maps joint velocities to end-effector spatial velocity. This enables precise, force-controlled manipulation tasks.

• Key Concept: Control laws are formulated directly in the 6D task space.
• Benefit: Provides decoupled control of linear and angular motion at the end-effector.

Calculating the joint forces/torques required to achieve a specific motion trajectory is vastly simplified. The Recursive Newton-Euler Algorithm uses spatial vectors to propagate velocities and accelerations inward from the base to the end-effector, then calculates forces outward. This is essential for model-based control and trajectory optimization.

• Process: Forward pass (kinematics) + Backward pass (dynamics).
• Output: The required actuator efforts for a given motion.

The spatial Jacobian is a 6xn matrix that maps n joint velocities to the 6D spatial velocity (linear + angular) of a link. It is constructed recursively and is fundamental for velocity kinematics, singularity analysis, and the aforementioned control methods. Its structure is more compact than separate linear and angular Jacobians.

• Construction: Built column-by-column from base to link.
• Use: Critical for differential kinematics and force transformation.

Modeling joints (revolute, prismatic, spherical) and loop constraints is natural with spatial algebra. Joint motion subspaces are defined using spatial vectors, and constraint forces are solved using spatial equations of motion. This is the foundation for simulating complex mechanical systems like vehicle suspensions or parallel manipulators.

• Key Concept: Each joint type has a characteristic motion subspace (S).
• Solver Integration: Constraints feed into the constraint solver of a physics engine.

A 6x6 matrix that compactly represents the mass, center of mass, and rotational inertia of a rigid body within the spatial vector framework. It is the key operator that relates spatial acceleration to spatial force.

• Structure: Combines the 3x3 inertia tensor (rotational), mass (linear), and first mass moments (cross terms) into a single, symmetric matrix.
• Transformation: Obeys specific coordinate transformation rules, allowing inertia to be expressed at any point on the body.
• Purpose: Enables the concise formulation of the Newton-Euler equations in 6D, forming the mass matrix in spatial dynamics.

A 6x6 matrix that transforms spatial vectors (like velocity, acceleration, and force) between two coordinate frames attached to a rigid body. It is the workhorse for changing the reference point and orientation of spatial quantities.

• Composition: Built from a 3x3 rotation matrix and the 3D vector from the old frame to the new frame.
• Dual Types: Exists in motion (transform) and force (transpose inverse) forms to correctly transform velocities/accelerations and forces, respectively.
• Critical Role: Fundamental for propagating motion and force across joints in an articulated mechanism using the Featherstone algorithm.

A 6x6 matrix that represents the effective inertia of a subtree of an articulated mechanism, as felt at a specific joint. It is a core recursive quantity in the Featherstone Algorithm.

• Dynamic Property: Not just the physical inertia of a single link, but the inertia that results from the motion of all descendant bodies.
• Recursive Calculation: Built from the spatial inertia of a link and the articulated body inertias of its children, projected through their joints.
• Use Case: Directly used to compute the joint acceleration resulting from an applied force, enabling the O(n) forward dynamics pass.

Special 6x6 matrices that define the cross product for spatial motion vectors and spatial force vectors. They are essential for expressing the Coriolis and centrifugal terms in the equations of motion.

• Two Variants: motion x motion → motion and motion x force → force. They maintain the correct Lie algebra structure for rigid body motion.
• Role in Dynamics: Appear in the spatial form of the Newton-Euler equations: *Spatial Force = Spatial Inertia * Spatial Acceleration + (Spatial Velocity) x (Spatial Inertia * Spatial Velocity)**.
• Implementation: Allow the dynamics of a complex system to be written in a compact, coordinate-free form that is ideal for algorithmic implementation.