Embodied AI Glossary中文04Mechanics & Kinematics
The physics behind robot motion: representing pose, forward and inverse kinematics, Jacobians, dynamics, and balance. · 168 terms
- 4.1Frames and pose13
- 4.2Ways to represent rotation19
- 4.3Links, joints, and mechanisms19
- 4.4Forward and inverse kinematics19
- 4.5Velocity and the Jacobian16
- 4.6Force, statics, and inertia13
- 4.7Dynamics and vibration21
- 4.8Contact, friction, and grasping21
- 4.9Legged balance and simplified models16
- 4.10Gait and walking11
4.1Frames and pose
The starting point: writing an object’s position and orientation as numbers with coordinate frames, then converting between frames.
- An idealized object where the distance between any two internal points never changes, no matter what force it feels.
- The number of independent variables needed to fully describe a system's configuration — often a robot's independently movable joints.
- A defined origin and set of axis directions used to describe positions and orientations.
- Right-Handed Frame & Axis Conventions右手坐标系与轴向约定Conventions for how x, y, z are arranged and which axis points up — inconsistent across software, and a frequent source of bugs.
- World Frame世界坐标系The single, unmoving reference frame shared by an entire scene, to which all other positions can be converted.
- Base Frame基坐标系The coordinate frame rigidly fixed to a robot's base, used as the reference for the arm's position and orientation.
- Body Frame机体坐标系A coordinate frame fixed to a robot's own body, moving and rotating along with it.
- Pose位姿An object's position plus orientation in space — six degrees of freedom in 3D.
- Rotation Matrix旋转矩阵A 3×3 orthogonal matrix that represents a 3D rotation; multiplying it by a vector rotates that vector.
- Special Orthogonal Group SO(3)特殊正交群 SO(3)The set of every rotation in 3D space, whose elements are orthogonal matrices with determinant 1.
- Converting a point's or pose's numeric value from one coordinate frame into another.
- A 4×4 matrix combining rotation and translation into one, describing one frame's pose relative to another.
- Gravity's straight-down direction expressed in the robot's own body frame, telling a policy how far the body is tilted.
4.2Ways to represent rotation
Expanding on ‘orientation’: Euler angles, quaternions, axis-angle, and 6D representations each have trade-offs, then on to Lie groups and screws.
- Euler Angles欧拉角A way of describing an object's orientation using three angles of rotation applied one after another about coordinate axes.
- Roll-Pitch-Yaw (RPY)横滚-俯仰-偏航角Orientation described by three rotation angles about the x, y, and z axes: roll, pitch, and yaw.
- For successive rotations, whether each step turns about the new, moving axes (intrinsic) or the original, fixed axes (extrinsic).
- Gimbal Lock万向节死锁When representing rotation with Euler angles, one angle reaching 90° makes two rotation axes coincide, losing a degree of freedom.
- Quaternion四元数A four-number representation of 3D rotation, the most common orientation format in robotics software.
- Quaternion Component Order (wxyz vs. xyzw)四元数分量顺序约定（wxyz / xyzw）Whether a quaternion's real part w is stored first or last — a convention that differs across software libraries.
- A way to smoothly blend between two rotations along the shortest arc, at a constant angular speed.
- Quaternion Double Cover四元数双倍覆盖The unit quaternions q and −q represent the exact same rotation, so every orientation corresponds to two quaternions.
- 6D Rotation Representation6D 旋转表示Representing a rotation with the first two columns of its rotation matrix — 6 numbers — a format well suited to neural-network learning.
- 9D Rotation Representation9D 旋转表示Lets a network output nine unconstrained numbers, then uses SVD to project them onto the nearest valid rotation matrix.
- The minimum angle still needed to turn one orientation into another — the standard way to measure rotation error.
- Representing a 3D rotation by which axis it turns around and by how large an angle.
- Rodrigues' Rotation Formula罗德里格斯公式Given a rotation axis and an angle, this formula computes the corresponding 3D rotation matrix directly, in closed form.
- A matrix that equals the negative of its own transpose; in 3×3 form it turns a cross product into matrix multiplication.
- A mathematical object that is both a group and a smooth manifold; rotations and poses in robotics are both Lie groups.
- Exponential Map指数映射Turns an axis-times-angle vector into a rotation or pose; the log map does the reverse.
- Screw Theory旋量理论Treats any rigid-body motion as a rotation about some axis combined with a translation along that same axis, like turning a screw.
- Plücker CoordinatesPlücker 坐标（Plücker 射线嵌入）Represents a line in 3D space with 6 numbers — a direction plus a moment — and is often used to encode camera rays per pixel.
- Dual Quaternion对偶四元数An 8-number algebraic tool that represents a 3D rotation and translation together in a single object.
4.3Links, joints, and mechanisms
From a single rigid body to a whole robot: how links and joints form a mechanism, and its degrees of freedom.
- Link连杆One of the rigid segments of a robot's body that joints connect together.
- Revolute Joint转动关节A joint that only lets two links rotate relative to each other about a fixed axis, giving 1 degree of freedom.
- Prismatic Joint移动关节A single-degree-of-freedom joint that lets two parts slide relative to each other along one straight axis, with no rotation.
- A three-degree-of-freedom joint that lets one part rotate about a fixed point in any direction, but not translate.
- Kinematic Pair运动副（转动副 / 移动副 / 低副 / 高副）A connection between two parts that stay in contact while allowed to move relative to each other, like a hinge, a slider, or meshing gears.
- Flexion/Extension and Abduction/Adduction屈伸与侧摆（外展/内收）Two basic joint motions: flexion/extension is bending and straightening, abduction/adduction is spreading apart and drawing together.
- Joint Limits关节限位The allowed range of motion for each joint, and more broadly the velocity and torque limits that go with it.
- A mechanism model made of rigid links strung together by joints, split into open chains and closed chains.
- Serial Mechanism串联机构A mechanism whose links connect end-to-end through joints from base to tip, with no closed loop.
- A mechanism connecting a base to an end platform through several kinematic chains at once — stiff and fast, but with a small workspace.
- Four-Bar Linkage四连杆机构Four bars joined end to end by four revolute joints into a closed loop, with just 1 degree of freedom.
- Grübler's FormulaGrübler 公式Computes how many degrees of freedom a mechanism has from its number of links, joints, and each joint's own freedom.
- An ankle joint where two motors, through linkages, jointly drive both pitch and roll instead of each controlling one axis alone.
- Kinematic Tree运动学树A tree structure with links as nodes and joints as edges, branching outward from a root to each end effector.
- A robot whose torso isn't bolted to the ground and instead has its own 6 degrees of freedom of free motion.
- The space of every possible pose a robot can be in, where each point represents one complete configuration.
- The smallest set of independent variables — such as each joint's angle — that uniquely describes a robot's overall configuration.
- Active DoF / Passive DoF主动自由度 / 被动自由度A degree of freedom is active when a motor drives it independently, and passive when it just follows a linkage or an external force.
- The motors can't directly produce acceleration in every direction, typically because there are fewer actuators than degrees of freedom.
4.4Forward and inverse kinematics
Once the structure is fixed, compute away: converting between joint angles and end-effector pose, plus modeling, reach, and calibration.
- Kinematics运动学The study of motion's pure geometry — position, velocity, acceleration — without regard to the forces that cause it.
- Joint Space关节空间The space described by a vector of all a robot's joint values — angles or displacements — used to describe its configuration.
- Task Space任务空间The space that directly describes what a task cares about, usually the end effector's position and orientation.
- The position and orientation of an arm's gripper or tool in space, usually given relative to the base frame.
- Tool Center Point工具中心点The point on the end-of-arm tool that actually does the work; the robot aims and moves this point, not the mounting flange.
- Computing the position and orientation of an arm's end effector in space, given all the joint angles.
- Working backward from a target end-effector position and orientation to find the joint angles that reach it.
- Workspace工作空间The region of positions, and sometimes orientations, that a robot's end effector can reach.
- The farthest distance a fully extended arm can reach from its base center — a basic spec for choosing a robot.
- A convention for modeling an arm where just four numbers per joint describe the transform between neighboring link frames.
- A variant of DH parameters that places each link's frame on the joint axis nearer the base, with a different transform order.
- Treats each joint as motion about a screw axis, and multiplies each joint's matrix exponential together to get the end-effector pose.
- An inverse-kinematics method that starts from an initial guess and repeatedly refines the joint angles to approach a target pose.
- An inverse-kinematics method that plugs a target end-effector pose into pre-derived formulas to get all joint angles directly.
- Spherical Wrist球形手腕A wrist structure where a manipulator's last three joint axes all intersect at a single point.
- Reachability Map / Inverse Reachability Map可达性地图 / 逆可达性地图A precomputed record of which end-effector poses an arm can reach; the inverse version looks up where to place the base to reach a target.
- Measuring the actual joint angle when a sensor reads zero, so the software model matches the physical robot.
- Measuring a robot's actual end-effector positions to correct the model's link lengths, joint zero offsets, and other geometric parameters.
- How far a robot's actual end-effector position ends up from the theoretical target position it was commanded to reach.
4.5Velocity and the Jacobian
Moving from position to velocity: the Jacobian converts joint velocity to end-effector velocity, and is where singularities and redundancy come from.
- The kinematics that relates joint velocities to end-effector velocity through the Jacobian matrix.
- Jacobian Matrix雅可比矩阵The matrix that converts joint speeds into end-effector velocity; it changes as the robot's posture changes.
- Twist运动旋量（速度旋量）Packs a rigid body's angular and linear velocity into one 6-dimensional vector that fully describes its instantaneous motion.
- Geometric vs. Analytical Jacobian几何雅可比与解析雅可比Both map joint velocity to end-effector velocity; they differ in whether orientation is expressed as angular velocity or as coordinate derivatives.
- An arm posture where the Jacobian matrix loses rank, so the end effector can't move in certain directions.
- Manipulability可操作度How freely an arm's end effector can move in each direction from its current pose, and how close it is to a singularity.
- The stand-in for a Jacobian inverse when it doesn't exist, giving the solution with the smallest error and least joint motion.
- Damped Least Squares阻尼最小二乘法Adds a damping term to the Jacobian inverse in numerical IK so the arm doesn't move wildly near singular poses.
- Uses the Jacobian's transpose instead of its inverse to iteratively solve IK — cheap to compute but slower to converge.
- Kinematic Redundancy运动学冗余Having more joints than a task strictly needs, so the same end-effector pose can be reached by infinitely many joint configurations.
- Null Space零空间Every input that a matrix maps to zero; for a redundant arm, the joint motions that don't change the end-effector pose at all.
- Swivel Angle臂型角（肘部自运动角）For a 7-DoF arm holding a fixed hand pose, the angle the elbow can still rotate through around the shoulder-wrist line.
- Solves for a robot's entire set of joint angles at once, so the hands, feet, and torso all reach their targets together.
- The equations relating the left and right wheel speeds of a two-wheeled base to its forward speed and turning rate.
- A constraint on velocity direction only, which can't be integrated into a position constraint — like a wheel that can't slide sideways.
- Jerk加加速度The rate of change of acceleration over time — the third time-derivative of position — measured in m/s³.
4.6Force, statics, and inertia
Geometry so far; now force: torque, wrenches, static equilibrium, and center of mass and inertia.
- Torque力矩The turning effect of a force on an object, equal to force times lever arm, measured in newton-meters.
- Payload负载The maximum mass a robot's end effector can reliably carry while working, usually given in kilograms.
- Wrench力旋量Packs a 3D force and a 3D torque into one 6-dimensional vector describing the full load acting on a rigid body.
- The 6×6 matrix that transforms a twist or a wrench from one coordinate frame into another.
- Statics静力学The branch of mechanics that studies how forces and torques balance on an object at rest or moving at constant velocity.
- A system is in static equilibrium exactly when the total work done by active forces over any allowed tiny virtual displacement is zero.
- Assuming motion is slow enough that inertial forces can be ignored, so every instant can be analyzed as a force balance.
- The mass-weighted average position of an object, which behaves as if all its mass were concentrated there.
- How hard an object is to spin up or slow down about a given axis, I = Σmr².
- Inertia Tensor惯性张量A 3×3 symmetric matrix describing how hard a rigid body is to rotate about any given axis.
- Gives the moment of inertia about any parallel axis from the moment of inertia about the center-of-mass axis: I = I_c + md².
- The set of values describing how heavy a rigid body is, where its center of mass sits, and how hard it is to rotate about each axis.
- A measure of an object's rotational “oomph”; for a rigid body spinning about an axis, it equals moment of inertia times angular velocity.
4.7Dynamics and vibration
Connecting force and motion: writing the equations of motion, solving them efficiently with recursive algorithms, then spring-damper vibration.
- Dynamics动力学The study of how forces and torques produce motion — how much acceleration a given torque causes.
- Rigid-Body Dynamics刚体动力学The study of how forces and torques cause acceleration and rotation in one or more connected rigid bodies.
- Multibody Dynamics多体动力学The study of how systems of rigid or flexible bodies, connected by joints, move under applied forces.
- Forward Dynamics正动力学Computing a robot's current acceleration, and hence its next motion, from its current state and the torques applied to it.
- Inverse Dynamics逆动力学Working backward from a desired motion (position, velocity, acceleration) to the torque each joint needs to produce.
- Newton-Euler Equations牛顿-欧拉方程A pair of dynamics equations describing a rigid body's translation (F = ma) and rotation (Euler's equation) together.
- Euler-Lagrange Equations拉格朗日方程A way of deriving a robot's equations of motion from kinetic and potential energy, without analyzing constraint forces one by one.
- Mass Matrix质量矩阵The matrix M(q) in a robot's dynamics equation that describes how hard each joint is to accelerate.
- Coriolis and Centrifugal Terms科里奥利力与离心力项The torque terms in a robot's equations of motion that depend on the square or product of joint velocities.
- Recursive Newton-Euler Algorithm递归牛顿-欧拉算法Sweeps velocity and acceleration outward from the base, then sweeps force back inward from the tip, to compute inverse-dynamics joint torques.
- An algorithm that efficiently computes the mass matrix by merging each joint's outboard links into one composite rigid body.
- An O(n) recursive algorithm that computes joint accelerations from joint torques — the classic solution to forward dynamics.
- Spatial Vector Algebra空间向量代数A notation that packs angular and linear quantities into 6-dimensional vectors for computing multi-body dynamics.
- Moving the robot through specific trajectories and using the torque data to work out each link's mass and inertia.
- How strongly an object or joint resists deforming — the same force causes less deformation the stiffer it is.
- Compliance柔顺性How readily a robot yields to an external force; numerically, the inverse of stiffness.
- Damping阻尼The effect that gradually dissipates a moving or vibrating system's energy, settling it down and slowing it to rest.
- Mass-Spring-Damper System质量-弹簧-阻尼系统The most basic vibrating system, made of a mass, a spring, and a damper, used as an approximation throughout control and contact modeling.
- The frequency at which a system oscillates on its own with no sustained outside force; matching this frequency causes resonance.
- A dimensionless number measuring how fast oscillation decays — it determines whether a system overshoots and rings before settling.
- Flexible Joint柔性关节A joint with noticeable elasticity between the motor and the link, so the motor's angle doesn't equal the link's actual angle.
4.8Contact, friction, and grasping
From the robot itself to the outside world: contact forces, friction, and collisions, then using them to analyze a stable grasp.
- The force two objects exert on each other when touching, split into a normal (pressing) part and a tangential (friction) part.
- Normal Force and Tangential (Shear) Force法向力与切向力（剪切力）The contact-force component perpendicular to a surface is normal force; the component parallel to it is tangential (shear) force.
- The ratio of maximum friction force to normal force, determining how easily a contact surface slips.
- Coulomb Friction库仑摩擦The classic friction model where friction force never exceeds the friction coefficient times normal force, independent of sliding speed.
- Slip打滑Relative sliding between contact surfaces, usually because the tangential force needed exceeds the friction limit.
- The cone-shaped set of every force a contact point can apply on an object without slipping.
- Friction Pyramid摩擦金字塔Approximates the round friction cone with a polyhedral pyramid, turning the friction constraint into linear inequalities.
- Torsional and Rolling Friction扭转摩擦与滚动摩擦Two more kinds of friction: one resists an object spinning in place, the other resists it rolling across a surface.
- Static Friction (Stiction) and Stribeck Effect静摩擦与 Stribeck 效应Getting something moving takes extra force to beat static friction first, and once it starts, friction actually drops before rising again.
- Viscous Friction黏性摩擦Friction proportional to relative speed — the faster something turns, the more resistance it meets.
- Impact碰撞冲击When two objects collide, a very brief but very large contact force that abruptly changes their velocities.
- The ratio of separation speed after a collision to approach speed before it — a measure of how bouncy the impact is.
- Grasp Taxonomy (Power Grasp vs. Precision Grasp / Pinch)抓取分类（强力抓取 / 精确抓取 / 捏取）A classification of hand grasps by how the hand contacts an object, split at the top level into power grasps and precision grasps.
- Antipodal Grasp对跖抓取A two-finger grasp where the contact points face each other and their connecting line lies inside both friction cones.
- A grasp where friction at the contact points can resist an external force or torque in any direction, so the object can't be pulled free.
- Form Closure形封闭Immobilizing an object purely through the geometry of contact points, with no reliance on friction at all.
- Contact Jacobian接触雅可比The matrix that maps joint velocities to contact-point velocities, and contact forces back into joint torques.
- Grasp Matrix抓取矩阵The linear map that combines each contact point's force into the total force and torque acting on the grasped object.
- Grasp Quality Metric抓取质量指标A single number scoring a grasp by how well it can resist outside disturbances, used to rank candidate grasps.
- Hand Synergies手部协同The many joints of a human hand tend to move together in a few fixed combinations, describable by a handful of synergy parameters.
- Fingers release and reposition one at a time while the rest keep a firm grip, rotating an object through a large angle.
4.9Legged balance and simplified models
Applying contact forces to the feet: how legged robots judge whether they’re stable, and the simplified models used to plan balance.
- Support Polygon支撑多边形The convex region enclosed by all of a robot's ground-contact points; balance criteria are measured against this boundary.
- Static Stability静态稳定The robot's center of mass projects straight down inside its support polygon, so it stays balanced even standing still.
- Balance where the center of mass can briefly leave the support region, recovered through continued motion and the next footstep.
- The force the ground pushes back onto a foot; legged robots stand, walk, and jump entirely by means of it.
- The single point where the ground's total supporting force on a foot effectively acts, used to judge balance.
- The point where the ground-reaction force produces no horizontal moment — the classic balance criterion for biped walking.
- Point Foot vs. Flat Foot点足 / 平足（足端形态）Whether a leg's foot meets the ground at a single point or across a flat sole determines whether ankle torque can hold balance.
- A simplified model of a robot as a pole on the ground balancing a point mass, used to analyze and control balance.
- A simplified model of bipedal walking that holds the center-of-mass height constant, used for fast gait planning.
- Cart-Table Model小车-桌子模型Models a biped as a cart rolling on a tabletop, used to connect center-of-mass motion to the zero moment point.
- The point on the ground where, if the robot steps right now, it would come to a complete stop.
- The part of a biped's center-of-mass motion that diverges exponentially; controlling it is enough to keep the robot balanced.
- Centroidal Moment Pivot质心力矩枢轴点The point where a line through the center of mass, parallel to the ground-reaction force, meets the ground.
- Centroidal Dynamics质心动力学Dynamics that track only how a robot's center of mass and overall momentum change under external forces.
- The matrix that maps a robot's full-body joint velocities into its center-of-mass linear and angular momentum.
- Single Rigid Body Dynamics Model单刚体动力学模型A simplified dynamics model that treats a whole legged robot as one rigid body and ignores the mass of its legs.
4.10Gait and walking
From standing to walking and running: gait phases and types, running and hopping models, and more natural, efficient ways to walk.
- Stance Phase / Swing Phase支撑相 / 摆动相The gait phase where a foot is on the ground bearing weight is stance; the phase where it swings through the air is swing.
- The part of a gait where every foot is off the ground at once and the body is airborne.
- Gait Cycle and Duty Factor步态周期与占空比The gait cycle is one full stride from touchdown to the next touchdown; the duty factor is the fraction spent on the ground.
- Pace Gait踱步步态A two-beat quadruped gait where both legs on the same side step together, alternating left and right.
- Bound Gait跳跃步态A running gait where a quadruped's two front legs land together, then the two rear legs land together, alternating front and back.
- Gallop Gait疾驰步态The fastest quadruped gait: all four feet touch down in sequence, slightly offset, with a phase where the whole body is airborne.
- Models running and hopping with a point mass on top of a single massless spring leg.
- A humanoid walks with its supporting leg nearly straight, rather than staying bent-kneed in a crouch the whole time.
- Passive Dynamic Walking被动动力学行走A style of biped walking with no motors, powered purely by gravity and the legs' natural swing down a gentle slope.
- Limit Cycle极限环An isolated closed periodic trajectory in a nonlinear system that nearby states get pulled toward and circle around.
- The energy used to move a unit of weight a unit of distance — a measure of how efficient walking or running is.