16-311-Q I NTRODUCTION TO R OBOTICS L ECTURE 4: (N ON ) H OLONOMIC R OBOTS W HEELED R OBOTS , K INEMATICS I NSTRUCTOR : G IANNI A. D I C ARO
R O B O T “ S PA C E S ” Representation and Robot design control • Components, Design, Geometry • Workpace • Stability, reliability • Task space • Maneuverability • Configuration space • Controllability • Task efficacy •Can the robot fully access its task space? ➔ Task e ffi cacy •Is the robot able to move between two feasible poses/configurations without any restrictions? ➔ Maneuverability, Task e ffi cacy •How di ffi cult, reliable is to control robot motion? ➔ Controllability 2
LOCOMOTION ➔ KINEMATICS VS. DYNAMICS Refers to the process of moving from one point to another, which Locomotion: requires the application of forces The study of motion (of a mass) through the direct modeling of Dynamics: the forces that cause it The study of motion without taking into consideration the forces Kinematics: that cause it. It is based on geometric relations, positions, velocities, and accelerations. Forward Kinematics: Inverse Kinematics: Use of kinematic equations to determine / Given the desired final configuration (of the predict the final configuration/pose of a e ff ectors/pose), make use of the kinematic robot based on the specification of the equations to determine the values of the values for the control variables (e.g., v , 𝞉 ) control variables that allow to achieve it. The specification of the entire movement of the robot in terms of Motion planning: its control variables to achieve the desired configurations in (s,t). 3
M O T I O N C O N T R O L A N D M O T I O N P R E D I C T I O N Posture prediction: Forward Kinematics Posture regulation: Inverse Kinematics Controls R Controls ? Pose o b ? (v, 𝞉 ) o Robot Robot t Robot Goal pose Feasible? Robot Controls ? Path following R Robot Robot o (geometry) b Path( s ) o t Feasible? Robot Controls ? Trajectory following R Robot Robot o (kinematics, time) b Path(s, t ) o t Feasible? 4
FORWARD KINEMATICS FOR A ROBOT ARM A robot arm with two links connected by revolute joints: determine the end-e ff ector position, X + Specification of initial angles Specification of the two joint angles and velocities ➔ Integration of equations 5
INVERSE KINEMATICS FOR A ROBOT ARM Given the desired position X of the end-effector, determine the values for the joint variables Usually the problem admits Solve the kinematic equations wrt the final pose multiple solutions 6
F O R WA R D K I N E M AT I C S F O R A M O B I L E R O B O T Given: the geometric parameters: number and type of wheels, wheel(s) radius, length of axes, … the initial conditions: pose and velocity and assigned the spinning speeds of each wheel: ϕ 1 , ˙ ˙ ϕ 2 a forward kinematic model aims to predict the robot’s generalized velocity (rate of pose change) in the global reference frame: ⇤ T = f ( l, r, θ , ˙ ˙ ˙ ⇥ ˙ ˙ ξ W = θ ϕ 1 , ˙ ϕ 2 ) x y once integrated over time using the initial conditions, the new robot pose can be then computed (predicted) Strategy: compute the contribution to motion of each wheel, in the local reference frame and apply the transformations equations …. 7
M O T I O N C O N T R O L F O R M O B I L E V S . A R M R O B O T S Mobile robot’s controllability (e ff ectors + structure + constraints + mass) defines feasible paths and trajectories in the robot’s workspace. Di ff erence between mobile and arm robots: position estimation (of end e ff ector, robot pose) in the world (inertial) reference frame {W} Mobile: It can span the entire Arm: Constrained workspace ➔ environment , no direct/obvious way to Measures of all intermediate joints + Kinematic equations measure its position instantaneously/ exactly ➔ Integrate motion over time + include uncertainties and errors (e.g., due to wheels slippage) {W} {W} Fixed wrt {W} ? It is a much harder task! 8
G E N E R A L I Z E D C O O R D I N AT E S , C O N F I G U R AT I O N S PA C E R 2 R 2 × S 1 q = (x,y) q = ( θ 1 , θ 2 ) q = (x,y , θ ) q = (x,y , z, ɸ , θ , Ψ ) q = (s ) q = (s, θ ) q = ( θ ) 9
G E N E R A L I Z E D C O O R D I N AT E S , C O N F I G U R AT I O N S PA C E q = (u,v ) With obstacles 10
G E N E R A L I Z E D C O O R D I N AT E S , C O N F I G U R AT I O N S PA C E q = (x,y , φ , θ ) q = (x,y , φ , θ , Ψ ) q = (x,y , z, l, θ , Ψ ) DOFs = #independent generalize coordinates Is the robot able to move between two feasible configurations without any restrictions? ➔ Maneuverability q = (x,y , φ , θ ) 11
M O T I O N A C T U AT O R S I M P O S E L I M I T S Two-moves car parking: no side-way motion No easy side-way motion in 3D A train robot moving forward/backward on a track can reach any point in its configuration space without limitations regarding the trajectories 12
L E T ’ S S TA RT F O R M A L I Z I N G T H E S E L I M I TAT I O N S … A geometric constraint imposes restrictions on the achievable configurations of the robot. It is based on a functional relation among (some subset of) the configuration variables f ( ξ , t) = 0 A kinematic constraint imposes restrictions on the achievable velocities of the robot. It is based on a functional relation among configuration variables and their derivatives g ( ξ , d ξ /d t, t ) = 0 v If g() is linear in the derivatives the constraint is said Pfa ffi an 13
H O L O N O M I C A N D N O N H O L O N O M I C C O N S T R A I N T S A geometric constraint is expressed through “positional” variables (e.g., ( α , β , φ 1 , φ 2 , x, y, θ , …)) and is said holonomic . A holonomic constraint limits the motion of the system to a manifold of the configuration space , depending on the initial conditions A kinematic constraint can be integrable , meaning that it can be expressed in a form: f ( ξ , t) = 0 where ξ is a vector of configuration variables, and it becomes a holonomic constraint. A kinematic constraint which is not integrable is said an non holonomic constraint, meaning that it is expressed through “derivatives of positional variables” (and cannot be integrated to provide a constraint in terms of positional variables). A non holonomic constraint does not limit the accessible configurations, but limits the paths that can be followed to reach them . A car-like vehicle is an example of non holonomic vehicle: all poses can be achieved in the configuration space, but the paths to reach them can be complex (e.g., parallel parking is not allowed) A sliding puzzle is also non holonomic! 14
N E E D F O R W O R K I N G I N T H E V E L O C I T Y S PA C E S The presence of non holonomic constraints forces to work in the terms of transformations on velocities rather than on positions In presence of non holonomic constraints, the di ff erential equations of motion are not integrable to the final position. For instance, in a wheeled robot, the measure of the traveled distance of each wheel is not su ffi cient to calculate the final position of the robot. One has also to know how this movement was executed as a function of time. 15
N E E D F O R W O R K I N G I N T H E V E L O C I T Y S PA C E S Forward kinematics: Transformation from configuration space to physical space Inverse kinematics: Transformation from physical space to configuration space In mobile robotics, due to (pervasive presence of) non holonomic constraints, usually we need to work with di ff erential (inverse) kinematics : Transformation between velocities instead of positions (v, 𝞉 ) (x,y, θ ) 16
D I F F E R E N T I A L K I N E M AT I C S M O D E L s(t) 17
R O B O T P O S E : R E P R E S E N TAT I O N A N D E V O L U T I O N 𝛐 I represents the pose of the robot wrt the inertial global reference {l}, while 𝛐 R is the pose in the local robot reference frame {R}. . . R and 𝜊 I represent the related velocities: 𝜊 rate of change of the pose in the respective reference frames Instantaneous rotation matrix (in the dt time for pose change calculation) Note: in previous lecture the notation R( 𝛊 ) was used for the inverse of this matrix 18
E X A M P L E O F P O S E T R A N S F O R M AT I O N The robot is aligned with Y I . I , the motion (pose velocity) in Note: It’s more useful to compute 𝜊 the global frame { I} from the motion in the local frame {R} (which is what the robot can directly control) 19
T Y P E S O F W H E E L S To derive kinematic equations, we will add motion constraints due to the physical characteristics of the wheels … Castor axle Wheel axle Contact \ point Steered standard Fixed standard Castor 3 DoF 2 DoF 3 DoF Rollers … Spherical Swedish 3 DoF 3 DoF 20
T Y P E S O F W H E E L S Fixed standard Steered standard / Orientable d Mecanum/Swedish Castor / Spherical O ff -centered orientable 21
L O C O M O T I O N ( E F F E C T O R S ) I S M O R E T H A N W H E E L S … Locomotion using wheels (man-made) Locomotion modalities in natural systems 22
L O C O M O T I O N ( E F F E C T O R S ) I S M O R E T H A N W H E E L S … ✦ Walk ✦ Run ✦ Fly ✦ Swim ✦ Dive ✦ Drive ✦ Jump ✦ Crawl ✦ Roll ✦ Slide ✦ Flow ✦ …. 23
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