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Abstract:
This lecture introduces wheeled mobile robots, starting with general ground locomotion concepts and the importance of wheels for versatility and safety. It covers fundamental balance principles, including the support polygon and statical balance, illustrated with bicycle and tricycle examples. Different wheel types (fixed, steerable, caster, mechanum) and common robot kinematic structures (differential drive, synchro drive, tricycle, car-like, omnidirectional) are detailed. The lecture explains mobility constraints, distinguishing between geometric (global) and kinematic (local) constraints, leading to the concepts of holonomic (e.g., train) and non-holonomic (e.g., car) systems. Kinematic models are derived from constraints, often expressed in Pfaffian form, using the unicycle and bicycle models as key examples and showing their equivalence to more complex structures like differential drive and car-like robots. Finally, two fundamental control problems are addressed using the unicycle model: trajectory tracking, solved via error transformation and approximate linearization around equilibrium, and regulation/parking, solved using a different control law whose stability is demonstrated using the Lyapunov stability criterion.
Wheeled Mobile Robots: Kinematics and Basic Control
00:00:03 Lecture Introduction: Overview of the third control course lecture focusing on wheeled mobile robots, their kinematic models (state-space representation), and common control problems.
00:00:36 Ground Locomotion: Discusses robots moving via ground contact (wheels, legs, crawlers). Highlights the advantages of wheels for navigating diverse environments and inherent safety compared to legged systems.
00:02:02 Robot Balance: Introduces key concepts for stability:
00:02:45 Support Polygon: The convex hull formed by the robot's ground contact points (wheels, feet).
00:03:23 Statical Balance: Achieved when the vertical projection of the robot's center of mass falls within the support polygon. Examples: Tricycle (triangle polygon, stable) vs. Bicycle (line polygon, unstable).
00:06:05 Types of Wheels: Describes the four main categories:
00:06:40 Fixed Wheels: Fixed orientation relative to the chassis (e.g., car rear wheels), 1 DOF (spinning).
00:07:27 Orientable/Steerable Wheels: Variable orientation for steering (e.g., car front wheels), 2 DOF (spinning, steering).
00:08:19 Caster Wheels: Steering axis offset from wheel center (e.g., shopping cart wheels), typically passive, self-aligning with motion direction.
00:09:52 Swedish/Mechanum Wheels: Omnidirectional wheels enabling movement in any direction, typically active, can introduce slippage.
00:10:25 Common Robot Kinematic Structures: Introduces typical configurations:
00:10:34 Differential Drive: Two independently actuated fixed wheels + one passive caster wheel for balance (common in vacuum robots).
00:12:01 Synchro Drive: Multiple steerable wheels actuated simultaneously for both rolling and steering (common in service robots).
00:13:18 Tricycle: Two fixed wheels on a common rear axle + one front steerable wheel. Requires a differential gear (00:15:05) for smooth turning.
00:15:59 Car-like: Two fixed rear wheels (common axle) + two steerable front wheels (common axle).
00:16:33 Omnidirectional (Caster-based): Uses multiple actuated caster wheels for omnidirectional movement; can exhibit sudden realignments.
00:18:04 Mobility Constraints: Explains limitations on robot movement:
00:18:43 Geometric vs. Kinematic: Geometric constraints limit reachable configurations (global), while kinematic constraints limit instantaneous velocities based on configuration (local).
00:21:16 Holonomic vs. Non-holonomic: Holonomic constraints are integrable kinematic constraints that imply geometric ones (restrict global & local mobility, e.g., train on tracks). Non-holonomic constraints are non-integrable kinematic constraints (restrict only local mobility, e.g., car).
00:24:35 Deriving Kinematic Models: Process of obtaining mathematical models:
00:25:00 From Kinematic Constraints: Models are derived directly from the kinematic constraints.
00:25:50 Pfaffian Form: Constraints are often expressible as A(q) * q_dot = 0.
00:26:43 Null Space Solution: The velocity vector q_dot must lie in the null space of the constraint matrix A(q), leading to models of the form q_dot = G(q) * u.
Definition: A single steerable wheel; state includes position (x, y) and orientation (theta).
Constraint & Model: Derived from the "pure rolling" (no sideways slip) constraint. The resulting model has driving velocity (u1) and steering velocity (u2) as inputs.
Equivalence (00:32:57): The simple unicycle model mathematically represents the behavior of differential drive and synchro drive robots.
Holonomy (00:39:37): Non-holonomic; cannot move sideways instantaneously but can reach any pose through maneuvering.
Definition: Fixed rear wheel, steerable front wheel; state includes position (x, y), chassis orientation (theta), and steering angle (phi). Tilting is ignored.
Constraint & Model: Derived from pure rolling constraints on both wheels. Model inputs are driving velocity and steering velocity.
Equivalence (00:39:01): Mathematically represents car-like robots (a statically stable version of the bicycle).
Goal: Make the robot follow a time-varying desired path (Xd(t), Yd(t)).
Approach: Define tracking error, transform error coordinates to the robot's frame, perform input transformation.
Linearization (00:47:52): Uses approximate linearization around the equilibrium point (zero error) to handle system non-linearities, valid locally.
Controller: Design a state feedback controller for the linearized system.
Limitation (00:53:50): The derived controller requires the desired trajectory to have non-zero velocity (persistent excitation); cannot be used for stopping/parking.
Goal: Drive the robot to a fixed target location (e.g., the origin) and stop, ignoring the final orientation for simplicity.
Approach: Define error in polar-like coordinates (distance and angle errors). Design a control law where driving velocity depends on distance error and steering velocity depends on pointing error.
Lyapunov Stability (00:59:13): Introduces the Lyapunov stability criterion to prove controller stability without explicit linearization or eigenvalue analysis. Requires finding a function V(x) (Lyapunov function, like energy) such that V(0)=0, V(x)>0 for x!=0, and V_dot(x)<0 along system trajectories.
Proof Outline (01:02:06): Choose a candidate Lyapunov function (quadratic form of errors), calculate its time derivative using system dynamics and the control law, show V_dot is negative definite with positive controller gains.
Result: The controller asymptotically stabilizes the robot at the target location.
01:06:56 Lecture Recap: Summarizes covered topics: robot architectures, wheels, kinematic models from constraints (unicycle, bicycle), control problems (tracking, parking), and key theoretical tools introduced (linearization, Lyapunov stability).