Omnidirectional Robot Steering, Ackermann and Swerve Kinematics Explained

A rigid body maintains constant distances between all points during motion. The velocity of any point P is v_P = v_O + ω × r_PO, requiring six independent parameters (three translational, three rotational) to fully describe motion. Acceleration follows a_P = a_O + α × r_PO + ω × (ω × r_PO). Absolute angular quantities measure rotation relative to inertial frames, while relative angular quantities measure rotation relative to moving bodies. Angular velocity vectors add according to vector rules, but scalar angles do not. When two bodies are rigidly connected, their relative angular velocity is zero; when relative angular velocity equals the negative of base angular velocity, net absolute angular velocity becomes zero.

This comprehensive section covers the foundational concepts of kinematics and velocity analysis in mechanical engineering. A kinematic link is two bodies connected with relative motion. A kinematic pair is when two links connect with relative motion. A kinematic chain consists of multiple kinematic pairs connected together. When one link in a kinematic chain is fixed, it becomes a mechanism. Inversion of a mechanism occurs when different links are fixed one by one, producing different mechanisms from the same kinematic chain. The three types of constrained motion are: completely constrained motion (motion in one direction), incompletely constrained motion (motion in multiple directions), and successfully constrained motion (motion completed by another link). Lower pairs involve surface contact, while higher pairs involve point contact including rolling, screw, and spherical pairs. Open pairs are in contact only by force, while closed pairs are in mechanical contact. A machine utilizes energy and converts it to useful work, while a structure transmits forces but not motion. Linear velocity is defined as the rate of change of linear displacement with respect to time. Angular velocity is defined as the rate of change of angular displacement with respect to time. The relationship between linear and angular velocity is given by v = rω. Velocity diagrams are constructed by drawing perpendicular lines from the crank center and connecting rod, then measuring the velocity vectors. The velocity triangle is used to find the velocity of the piston.

This section introduces the kinematics of rigid bodies, covering absolute velocity, relative velocity, and general plane motion. The instructor explains how to analyze complex mechanical systems like vibrating bridges by decomposing motion into translation and rotation. The core concept is that general plane motion can be analyzed as a superposition of translation (where all points move with the same velocity) and rotation (where points move in circular paths about a fixed reference point). The fundamental velocity relationship between two points on the same rigid body is established: v_B = v_A + ω × r_B/A, where the relative velocity is obtained through the cross product of angular velocity and the position vector.

This comprehensive section covers the foundational concepts of rigid body kinematics. The instructor distinguishes rigid body kinematics from particle kinematics, explaining that rigid bodies have size and shape, allowing different points to move differently. Three fundamental types of motion are introduced: translational motion where all points move identically, rotational motion where points move different distances based on their distance from the rotation axis, and general plane motion which combines both translation and rotation. Angular velocity (omega) is introduced as the rate of change of angular position, measured in radians per second, and is fundamental to rigid body kinematics. The instructor explains that omega relates rotational motion to translational motion through the relationship v = ωr. Rolling motion is presented as a special case where a body rolls without slipping, meaning the contact point with the ground has zero velocity. The mathematical framework for relative velocity analysis is developed, including the formula v_B/A = ω × r_B/A, where the direction is perpendicular to the line connecting the points and determined by the rotation direction. The instructor emphasizes that understanding the geometry and rotation direction is essential for correctly determining velocity components.

Rigid Body Dynamics (RBD) studies the motion of rigid bodies. A rigid body is defined as a body whose shape, size, and geometry remain fixed and constant. For a rigid body, the distance between any two particles must remain constant, and the velocity of separation along the line connecting two particles must be zero. Only velocity perpendicular to the line connecting particles is allowed. If one particle performs circular motion with respect to another, the angular velocity must be the same for all particles. The relative angular velocity between any two particles must be equal. For a disk rotating with angular velocity ω, a particle at distance r from the center has linear velocity v = rω. The angular velocity of this particle with respect to the center is ω. This demonstrates that all particles in a rigid body have the same angular velocity regardless of their distance from the axis of rotation.

Skid steer machines use a differential drive system where each wheel has its own axle, allowing independent speed control for turning. When going straight, both chains travel at the same speed. When turning, one chain slows to one-third speed while the other maintains full speed, causing the machine to pivot around the slower side. In high gear, one side turns full speed while the other turns at one-third speed; in low gear, one side turns full speed while the other stops completely, enabling sharp turns.

Skid steering uses four fixed wheels with independent velocity control. Unlike differential drive, the wheel axes never intersect, meaning there is no instantaneous center of curvature. This causes wheels to skid on the ground, creating bumpy motion and potential floor damage. The system is good for rough outdoor environments where gentle motion is not required. Turning is achieved by applying different velocities to left and right wheels, but the lack of ICC satisfaction makes it less suitable for indoor environments or applications requiring smooth motion.

A skid steer operates by locking up one side of the wheels while powering the other side to turn. This creates a pivoting motion where the machine rotates around the locked wheels. The steering is controlled by a brain (controller) that tells the motors which side to operate, enabling the machine to turn by differentially powering each side.

Skid steer loaders feature a distinctive drive system where wheels on one side move in one direction while wheels on the opposite side move in the opposite direction, enabling tight turning capabilities. Traditional skid steer controls use two levers positioned on either side of the operator's seat. However, some modern models offer alternative configurations, such as forward, reverse, left, and right shuttle controls, which can be more intuitive for operators accustomed to different control schemes. Boom controls are typically managed through a right-hand joystick with proportional control, while auxiliary functions like muck grab operation are controlled via toggle switches.

Skid steer loaders use a unique steering mechanism where each wheel can move independently. The left and right wheels can move in different directions (left wheel forward/backward, right wheel forward/backward), allowing the machine to spin on a dime. This provides excellent maneuverability but makes it difficult to see backwards, requiring careful operation.

This section covers vector addition and resultant forces: (1) When two velocities act on an object, the resultant is found by vector addition: same direction (add), opposite direction (subtract), perpendicular (use Pythagorean theorem); (2) When two vectors act at an angle θ, the magnitude of the resultant is: R² = A² + B² - 2AB cos(θ); (3) For perpendicular vectors (θ=90°), R² = A² + B² (Pythagorean theorem); (4) For θ=60°, cos(60°)=0.5, so R² = A² + B² - AB.

The resultant velocity of an object is found by vector addition of its own velocity and the velocity of the medium (such as wind or water current). This is done using the 'head-to-tail' method (also called the 'bundinha na cabecinha' rule), where you place the tail of one vector at the head of another. The resultant vector connects the starting point to the endpoint. When the medium's velocity has a component in the same direction as the object's motion, the resultant is larger; when it opposes, the resultant is smaller.

To find the resultant of two vectors, resolve them into components along the direction of one vector. If the resultant W makes angle θ with velocity u, then W cosθ = u + v cosα. Given W cosθ = v, we get u cos0 + v cosα = v, so cosα = (v - u)/v. Then substitute into the parallelogram law to find the resultant magnitude. This component method is useful when the resultant direction is given.

To find an object's velocity relative to a stationary observer, add the object's velocity relative to its moving frame plus the frame's velocity relative to the observer. When velocities are in the same direction, simply add them (e.g., 5 m/s + 1 m/s = 6 m/s). When in opposite directions, add algebraically with signs (e.g., 5 m/s + (-1 m/s) = 4 m/s). For perpendicular directions, use the Pythagorean Theorem: C² = A² + B². This gives both magnitude and direction of the resultant velocity vector.

Vectors have magnitude and direction, represented by arrows where length shows magnitude and head shows direction. Addition: link head of first to tail of second; resultant connects tail of first to head of second. Subtraction: reverse direction to create negative vector, then add. Resolution: horizontal component = F cosθ, vertical component = F sinθ. Vector notation: F = Fh i + Fv j = F cosθ i + F sinθ j. Magnitude = √(Fh² + Fv²), direction = tan⁻¹(Fv/Fh). Example: aircraft velocity 200 km/h north with wind 80 km/h from 30° west of north yields resultant velocity ≈137 km/h at 17° east of north using cosine rule and sine rule. These techniques enable analysis of combined effects in physics problems involving multiple vector contributions.