Movement of robots and introduction to kinematics of

Movement of robots and introduction to kinematics of robots

• Kinematics: constraints on getting around the environment kinematics The effect of a robot’s geometry on its motion. wheeled platforms from sort of simple to sort of difficult manipulator modeling

Effectors and Actuators • An effector is any device that affects the environment. • A robot's effector is under the control of the robot. • Effectors: – legs, – wheels, – arms, – fingers. • The role of the controller is to get the effectors to produce the desired effect on the environment, – this is based on the robot's task.

Effectors and Actuators • An actuator is the actual mechanism that enables the effector to execute an action. • Actuators typically include: include – – electric motors, hydraulic cylinders, pneumatic cylinders, etc. • The terms effector and actuator are often used interchangeably to mean "whatever makes the robot take an action. " • This is not really proper use: – Actuators and effectors are not the same thing. – And we'll try to be more precise.

Review: degrees of freedom • Most simple actuators control a single degree of freedom, • i. e. , a single motion (e. g. , up-down, left-right, in-out, etc. ). – A motor shaft controls one rotational degree of freedom, for example. – A sliding part on a plotter controls one translational degree of freedom. • How many degrees of freedom (DOF) a robot has is very important in determining how it can affect its world, – and therefore how well, if at all, it can accomplish its task. • We said many times before that sensors must be matched to the robot's task. • Similarly, effectors must be well matched to the robot's task also. When you design a robot your first task is decide the number of DOF and the geometry.

DOF • In general, a free body in space as 6 DOF: – three for translation (x, y, z), – three for orientation/rotation (roll, pitch, and yaw). • You need to know, for a given effector (and actuator/s): – how many DOF are available to the robot, – how many total DOF any given robot has. • If there is an actuator for every DOF, then all of the DOF are controllable. • Usually not all DOF are controllable, which makes robot control harder. To demonstrate, use a pen in your hand…. .

DOF and controllable DOFs • A car has 3 DOF: – position (x, y) and – orientation (theta). • But only 2 DOF are controllable: – driving: through the gas pedal and the forward-reverse gear; – steering: through the steering wheel. • Since there are more DOF than are controllable, there are motions that cannot be done. • Example of such motions is moving sideways (that's why parallel parking is hard).

One Minute Test • How many degrees of freedom does your hand have, with your forearm fixed in position? • (Hint: It’s not 6) Answer on next slide

Degrees of Freedom in Hand Part Wrist Palm Fingers Thumb Total Do. F Comment 2 1. Side-to-side 2. Up-down 1 1. Open-close a little 4*4 1. 2. 23 2 @ base (Up-down & side-to-side) 1 @ each of two joints 2 @ base (attached to wrist) 2 @ visible joints

Total and Controllable DOFs • Kinematics: y wheeled platforms vs. robot arms (or legs) x VL VR • We need to make a distinction between what an actuator does (e. g. , pushing the gas pedal) and what the robot does as a result (moving forward). • A car can get to any 2 D position but it may have to follow a very complicated trajectory. • Parallel parking requires a discontinuous trajectory with respect to the velocity. • It means that the car has to stop and go.

Definition of a HOLONOMIC robot • When the number of controllable DOF is equal to the total number of DOF on a robot, the robot is called holonomic. (i. e. the hand built by Uland Wong). Holonomic <= > Controllable DOF = total DOF

DOF versus types of robots Non-Holonomic <= > Controllable DOF < total DOF • If the number of controllable DOF is smaller than total DOF, the robot is non-holonomic. • If the number of controllable DOF is larger than the total DOF, the robot is redundant. (like a human hand, we did not build such robot yet) Redundant <= > Controllable DOF > total DOF

DOF for animals • A human arm has 7 DOF: – 3 in the shoulder, – 1 in the elbow, – 3 in the wrist • All of which can be controlled. • A free object in 3 D space (e. g. , the hand, the finger tip) can have at most 6 DOF! • So there are redundant ways of putting the hand at a particular position in 3 D space. • This is the core of why robot manipulation is very hard! One minute test!

JAPAN HONDA AND SONY ROBOTS • Pino, a 70 -centimeter (2 -foot)-tall and 4. 5 -kilogram (9 -pound) humanoid robot designed by Japan Science and Technology Corporation in Tokyo which can walk on its legs and respond to stimulation through a sensor, shakes hand with Malaysia's Prime Minister Mahathir Mohamad during the opening of the Expo Science & Technology 2001 in Kuala Lumpur, Malaysia, Monday, July 2, 2001. (AP Photo/Andy Wong) Question: how many DOF?

98 degrees (of freedom) • This is in any case simplified

Manipulation • In locomotion (mobile robot), the body of the robot is moved to get to a particular position and orientation. • In contrast - a manipulator moves itself – typically to get the end effector (e. g. , the hand, the fingertip) – to the desired 3 D position and orientation. • So imagine having to touch a specific point in 3 D space with the tip of your index finger; – that's what a typical manipulator has to do.

Issues in Manipulation • In addition: manipulators need to: – grasp objects, – move objects. – But those tasks are extensions of the basic reaching discussed above. • The challenge is to get there efficiently and safely. • Because the end effector is attached to the whole arm, we have to worry about the whole arm: – the arm must move so that it does not try to violate its own joint limits, – it must not hit itself or the rest of the robot, or any other obstacles in the environment.

Manipulation - Teleoperation • Thus, doing autonomous manipulation is very challenging. • Manipulation was first used in tele-operation, where human operators would move artificial arms to handle hazardous materials. • Complicated duplicates of human arms, with 7 DOF were built. • It turned out that it was quite difficult for human operators to learn how to tele-operate such arms

Manipulation and Teleoperation: Human Interface • One alternative today is to put the human arm into an exo-skeleton, in order to make the control more direct. • Using joy-sticks, for example, is much harder for high DOF. Exo-skeletons used in “Hollywood Robotics”

Why is using joysticks so hard? • Because even as we saw with locomotion, there is typically no direct and obvious link between: – what the effector needs to do in physical space – and what the actuator does to move it. • In general, the correspondence between actuator motion and the resulting effector motion is called kinematics. • In order to control a manipulator, we have to know its kinematics: – 1. what is attached to what, – 2. how many joints there are, – 3. how many DOF for each joint, – etc.

Basic Problems for Manipulation • Kinematics · Given all the joint angles - where is the tip ? • Inverse Kinematics · Given a tip position - what are the possible joint angles ? • Dynamics · To accelerate the tip by a given amount how much torque should a particular joint motor put out ?

Kinematics versus Inverse Kinematics • We can formalize all of this mathematically. – To get an equation which will tell us how to convert from, say, angles in each of the joints, to the Cartesian positions of the end effector/point is called: • computing the manipulator kinematics – The process of converting the Cartesian (x, y, z) position into a set of joint angles for the arm (thetas) is called: Something for lovers of math and • inverse kinematics. programming! Publishable! LISP

Links and Joints Links Joints: End Effector 2 DOF’s Robot Basis

Joints. • Joints connect parts of manipulators. • The most common joint types are: • Prismatic Link – revolute link (rotation around a fixed axis) – prismatic link (linear movement) These joints provide the DOF for an effector.

Joints. Revolute Link

Homogeneous Coordinates • Homogeneous coordinates: embed 3 D vectors into 4 D by adding a “ 1” • More generally, the transformation matrix T has the form: a 11 a 12 a 13 b 1 a 22 a 23 b 2 a 31 a 32 a 33 b 3 c 1 c 2 c 3 sf It is presented in more detail on the WWW!

Terms for manipulation • • Links and joints End effector, tool Accuracy vs. Repeatability Workspace Reachability Manipulability Redundancy Configuration Space

Direct Kinematics Where is my hand? Direct Kinematics: HERE!

Direct Kinematics • Position of tip in (x, y) coordinates

Direct Kinematics Algorithm 1) Draw sketch 2) Number links. Base=0, Last link = n 3) Identify and number robot joints 4) Draw axis Zi for joint i 5) Determine joint length ai-1 between Zi-1 and Zi 6) Draw axis Xi-1 7) Determine joint twist i-1 measured around Xi-1 8) Determine the joint offset di 9) Determine joint angle i around Zi 10+11) Write link transformation and concatenate

Kinematic Problems for Manipulation • Reliably position the tip - go from one position to another position • Don’t hit anything, avoid obstacles • Make smooth motions – at reasonable speeds and – at reasonable accelerations • Adjust to changing conditions – i. e. when something is picked up respond to the change in weight

Inverse Kinematics How do I put my hand here? IK: Choose these angles!

Why is using inverse kinematics so hard? • Inverse kinematics is computationally intense. · functions are nonlinear and complex , especially for higher dimensions than 2 • Difficult to visualize • Large number of inverse kinematics solutions due to redundancy • Large computational burden • And the problem is even harder if the manipulator (the arm) is redundant. • Manipulation involves: · trajectory planning (over time) · inverse kinematics · inverse dynamics · dealing with redundancy

Direct versus Inverse Kinematics • Direct Kinematics · x = L 1*cos(t 1) + L 2*cos(t 1+t 2) · y = L 1* sin(t 1) + L 2*sin(t 1+t 2) · Given the joint angles t 1 and t 2 we can compute the position of the tip (x, y) • Inverse Kinematics · Given x and y we can compute t 1 and t 2 · t 2 = acos[(x^2 + y^2 - L 1^2 - L 2^2)/(2*L 1*L 2)] · This gives us two values for t 2, now one can compute the two corresponding values of t 1. See next slide

Inverse Kinematics

One of many problems: There may be multiple solutions Elbow down - Elbow up

A Dynamic Simulator = kinematics + force modeling Building blocks of • masses • springs • “muscles” www. sodaconstructor. com

Wheeled Robots ~ 1. 5 cm to a side temperature sensor & two motors travels 1 inch in 3 seconds untethered !!

Other mini machines Pocketbot 55 mm dia. base radio unit Khepera linear vision gripper “Cricket” Accessorize! video

Kinematics of Differential drive

Kinematics of Differential drive Differential Drive is the most common kinematic choice All of the miniature robots… Pioneer, Rug warrior - difference in wheels’ speeds determines its turning angle Questions (forward kinematics) Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL VR Are there any inherent system constraints? 1) Specify system measurements 2) Determine the point (the radius) around which the robot is turning. 3) Determine the speed at which the robot is turning to obtain the robot velocity. 4) Integrate to find position.

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y VL 2 d VR (assume a wheel radius of 1) x

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. VL 2 d VR (assume a wheel radius of 1) x

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y VL 2 d x 2) Determine the point (the radius) around which the robot is turning. - to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles - each wheel must be traveling at the same angular velocity VR ICC (assume a wheel radius of 1) “instantaneous center of curvature” = angular velocity

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y w VL 2 d x 2) Determine the point (the radius) around which the robot is turning. - to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles - each wheel must be traveling at the same angular velocity around the ICC VR ICC (assume a wheel radius of 1) “instantaneous center of curvature”

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. - each wheel must be traveling at the same angular velocity around the ICC w VL x 2 d 3) Determine the robot’s speed around the ICC and its linear velocity VR ICC R robot’s turning radius (assume a wheel radius of 1) w(R+d) = VL w(R-d) = VR

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. - each wheel must be traveling at the same angular velocity around the ICC w VL x 2 d VR ICC 3) Determine the robot’s speed around the ICC and then linear velocity “instantaneous center of curvature” ICC w(R+d) = VL R w(R-d) = VR robot’s turning radius Thus, (assume a wheel radius of 1) w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL )

Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. - each wheel must be traveling at the same angular velocity around the ICC w VL x 2 d 3) Determine the robot’s speed around the ICC and then linear velocity VR ICC w(R+d) = VL R w(R-d) = VR robot’s turning radius Thus, w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) So, the robot’s velocity is V = w. R = ( VR + VL ) / 2

Kinematics of Differential drive 4) Integrate to obtain position y Vx = V(t) cos( (t)) w(t) Vy = V(t) sin( (t)) (t) VL x 2 d VR ICC Vx “instantaneous center of curvature” ICC R(t) robot’s turning radius with w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) What has to happen to change the ICC ? V = w. R = ( VR + VL ) / 2

Kinematics of Differential drive 4) Integrate to obtain position y Vx = V(t) cos( (t)) w(t) Thus, VL x(t) = x 2 d Vy = V(t) sin( (t)) y(t) = VR ∫ V(t) cos( (t)) dt ∫ V(t) sin( (t)) dt ∫ w(t) dt ICC R(t) robot’s turning radius with w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) V = w. R = ( VR + VL ) / 2

Kinematics of Differential drive Velocity Components y Vx = V(t) cos( (t)) speed Vy = V(t) sin( (t)) w(t) Thus, VL x(t) = V(t) cos( (t)) dt x 2 d y(t) = V(t) sin( (t)) dt (t) = VR w(t) dt ICC Kinematics R(t) robot’s turning radius with w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) What has to happen to change the ICC ? V = w. R = ( VR + VL ) / 2

Kinematics of Synchro drive

Kinematics of Synchro drive Nomad 200 wheels rotate in tandem and remain parallel all of the wheels are driven at the same speed Where is the ICC ?

Kinematics of Synchro drive Nomad 200 ICC at y wheels rotate in tandem and remain parallel all of the wheels are driven at the same speed w Vrobot = Vwheels wrobot = wwheels velocity x Vwheels (t) = w(t) dt x(t) = Vwheels(t) cos( (t)) dt position y(t) = Vwheels(t) sin( (t)) dt simpler to control, but. . .

Kinematics of Synchro drive Nomad 200 wheels rotate in tandem and remain parallel all of the wheels are driven at the same speed Question (forward kinematics) Given the wheel’s velocities or positions, what is the robot’s velocity/position ? Are there any inherent system constraints? 1) Choose a robot coordinate system 2) Determine the point (the radius) around which the robot is turning. 3) Determine the speed at which the robot is turning to obtain the robot velocity. 4) Integrate to find position.

Lego Synchro this light sensor follows the direction of the wheels, but the RCX is stationary also, four bump sensors and two motor encoders are included But how do we get somewhere? more difficult to build.

Inverse Kinematics Key question: Given a desired position or velocity, what can we do to achieve it? y x VL (t) VR(t) starting position final position

Inverse Kinematics Key question: Given a desired position or velocity, what can we do to achieve it? y x VL (t) VR(t) starting position final position

Inverse Kinematics Key question: Given a desired position or velocity, what can we do to achieve it? y x VL (t) VR(t) starting position final position

Inverse Kinematics Key question: Given a desired position or velocity, what can we do to achieve it? y Need to solve these equations: x = V(t) cos( (t)) dt y = V(t) sin( (t)) dt x VL (t) = w(t) dt w = ( VR - VL ) / 2 d V = w. R = ( VR + VL ) / 2 VR(t) starting position final position for VL (t) and VR(t). There are lots of solutions. . .

Inverse Kinematics Key question: Given a desired position or velocity, what can we do to achieve it? y Finding some solution is not hard, but finding the “best” solution is very difficult. . . x VL (t) VR(t) starting position final position • quickest time • most energy efficient • smoothest velocity profiles VL (t) t It all depends on who gets to define “best”. . .

Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time Differential Drive y x VL (t) VR(t) starting position final position

Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y -VL (t) = VR (t) = Vmax x VL (t) VR(t) starting position final position

Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y -VL (t) = VR (t) = Vmax (2) drive straight until the robot’s origin coincides with the destination x VL (t) = VR (t) = Vmax VR(t) starting position final position

Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y -VL (t) = VR (t) = Vmax (2) drive straight until the robot’s origin coincides with the destination x VL (t) = VR (t) = Vmax VR(t) starting position (3) rotate again in order to achieve the desired final orientation final position -VL (t) = VR (t) = Vmax VL (t) VR (t) t

Inverse Kinematics of Synchro Drive Usual approach: decompose the problem and control only a few DOF at a time Synchro Drive y (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. V(t) w(t) = wmax (2) drive straight until the robot’s origin coincides with the destination x V(t) = Vmax w(t) (3) rotate again in order to achieve the desired final orientation final position starting position w(t) = wmax sometimes it’s not so easy to isolate one or two DOF. . .

Other wheeled robots tricycle drive Ackerman drive Doubly-steered bicycle one more -- that roaming desk one more -- that roaming tatami mat (holonomic) & the XR 4000

Kinematics of Tricycle drive • back wheels tag along. . . Mecos tricycle-drive robot • front wheel is powered and steerable

Four-wheel Steering The kinematic challenges of parallel parking: • wheels have limited turning angles VFL • no in-place rotation VFR VBL VBR • small space for parking and maneuvers

Ackerman Steering • Similar to a tricycle-drive robot L R y VFL VFR g VBL d d VBR x r ICC r = g +d tan( R) wg sin( R) = VFR determines w

Ackerman Steering • Similar to a tricycle-drive robot L R y VFL VFR g r = g +d tan( R) wg sin( R) = VFR determines w The other wheel velocities are now fixed! VBL d wg sin( L) d VBR VFL L = tan-1(g / (r + d)) x r = w(r - d) = VBR w(r + d) = VBL ICC But this is just the cab. . .

The Big Rigs Applications 5 link trailer 2 controlled angles Parking two trailers

nonholonomicity All of the robots mentioned share an important (if frustrating) property: they are nonholonomic. - makes it difficult to navigate between two arbitrary points - need to resort to techniques like parallel parking

nonholonomicity All of the robots mentioned share an important (if frustrating) property: they are nonholonomic. - makes it difficult to navigate between two arbitrary points - need to resort to techniques like parallel parking By definition, a robot is nonholonomic if it can not move to change its pose instantaneously in all available directions. i. e. , the robot’s differential motion is constrained. Synchro Drive two DOF are freely controllable; the third is inaccessible

Holonomic Robots Navigation is simplified considerably if a robot can move instantaneously in any direction, i. e. , is holonomic. Omniwheels Mecanum wheels tradeoffs in locomotion/wheel design if it can be done at all. . .

Holonomic Robots Nomad XR 4000 Killough’s Platform synchro drive with offsets from the axis of rotation

Holonomic hype Holonomic Hype “The People. Bot is a highly holonomic platform, able to navigate in the tightest of spaces…”

Holonomic Hype Discover ‘ 97 -- Top 10 Innovation Sage -- a museum tour guide

Robot Manipulators Is this robot holonomic ? A robot holonomic if it can move instantaneously in any direction.

Robot Manipulators Is this robot holonomic ? No - it can’t move at all Yes - its end effector (a point) can translate instantaneously in the x or y directions Maybe - actually, in some cases the end effector is constrained. . .

Robot Manipulators Is this robot holonomic ? No - it can’t move at all Yes - its end effector (a point) can translate instantaneously in the x or y directions Maybe - actually, in some cases the end effector is constrained. . . Holonomic or not, the kinematics are vital to using a robot limb. . . Joint Angles Kinematics Useful Tasks

Robot Manipulators Forward kinematics -- finding Cartesian coordinates from joint angles • start by finding the position relationships, then velocity

Inverse Kinematics Inverse kinematics -- finding joint angles from Cartesian coordinates

Types of Manipulators Basic distinction: what kinds of joints extend from base to end. “RR” or “ 2 R” “PR” arm All manipulators can be represented as chains of P (prismatic) and R (rotational) joints.

Prismatic Joints Ninja Ambler tomato harvester

Challenges 1. Modeling many degrees of freedom 2. No closed-form solution guaranteed for the inverse kinematics. 3. Trajectory generation under nonholonomic constraints 4. Navigating with obstacles obstacle goal

Challenges Multiple solutions (or no solutions) for a task.

Challenges Multiple solutions (or no solutions) for a task.

Opportunities • Direct kinematic/ inverse kinematic modeling - is the basis for control of the vast majority of industrial robots. • Accurate (inverse) kinematic models are required in order to create believable character animations how would these things bike?

Configuration Space To get from place A to place B, we need a standardized notion of “place” : Configuration Space is a space representing robot pose. The dimensionality of C. S. is equal to the robot’s degrees of freedom. Examples: • The Nomad robot (discounting orientation) has a planar configuration space representing the (x, y) coordinates of the robot’s center. • The Nomad robot including orientation … • The 2 R manipulator depicted earlier. . . topological properties

To be discussed: Impinging on robots’ space: the next (but not final) frontier Getting from point A to point B robot navigation via path planning full-knowledge techniques insect-inspired algorithms

Perspective If your robot doesn’t do what you want. . . … you can always change what you’re looking for.

Dynamics of a one link arm • This differential equation can be solved to figure out what acceleration results from a particular given torque.

Control Techniques • P, PD, PID We will illustrate · simple, easy to implement • Impedance control · force and position • Advanced control techniques · robust, · sliding mode, · nonlinear etc. , · neural network based, · fuzzy logic based · etc.

PD Control of a One-Link Arm

PD Control of a One-Link Arm solution

Research Issues in Manipulators • Manipulators are well studied • Lots of hard problems (we’ve barely scratched the surface) • Modern techniques involve trying to use: – some of the kinematics – some of the dynamics of manipulators – sophisticated control theory – some learning

Navigation and Motion Planning 1 very little freedom c 1 c 2 • There is a similarity of planning movement of a hand of a mobile robot

Vertical Strip Cell Decomposition

Manipulation - Challenge for roboticists! • This is a challenging area of robotics. – We will cover it briefly in several lectures next quarter • Manipulators are effectors. • Joints connect parts of manipulators. • The most common joint types are: – rotary (rotation around a fixed axis) – prismatic (linear movement) • These joints provide the DOF for an effector, so they are planned carefully - kicking a ball in hexapod soccer?

More Challenges for roboticists! • Robot manipulators can have one or more of each of those joints. • Now recall that any free body has 6 DOF; – that means in order to get the robot's end effector to an arbitrary position and orientation, – the robot requires a minimum of 6 joints. • As it turns out, the human arm (not counting the hand!) has 7 DOF. – That is sufficient for reaching any point with the hand, – It is also redundant, meaning that there are multiple ways in which any point can be reached.

More Challenges for roboticists! • This is good news and bad news; – the fact that there are multiple solutions means that there is a larger space to search through to find the best solution. • Now consider end effectors. • They can be: – simple pointers (i. e. , a stick), – simple 2 D grippers, – screwdrivers for attaching tools (like welding guns, sprayer, etc. ), – or can be as complex as the human hand, hand with variable numbers of fingers and joints in the fingers.

Reaching and Grasping • Food for thought: how many DOF are there in the human hand? • Problems like reaching and grasping in manipulation constitute entire subareas of robotics and AI. • Issues include: – finding grasp-points (centers of gravity - COG, friction, etc. ); – force/strength of grasp; – compliance (e. g. , in sliding, maintaining contact with a surface); – dynamic tasks (e. g. , juggling, catching).

Advanced Manipulation • Other types of manipulation researched: – carefully controlling force, as in grasping fragile objects and maintaining contact with a surface (so-called compliant motion). • Dynamic manipulation tasks: – juggling, – throwing, – catching, etc. , are already being demonstrated on robot arms.

Problems to solve. • 1. Draw kinematics models of various animals and calculate the total DOFs. • 2. Give examples (drawings) of holonomic, non-holonomic and redundant mobile robots that you can build using standard components that you can find in the lab. • 3. Compare the kinematics of differential drive, the synchro drive and the four wheel steering. • 4. Give examples (drawings) of holonomic, non-holonomic and redundant robot arms that you can build using standard components that you can find in the lab.

Next Time(s) Inverse kinematics: what we would really like to know. . . Examining robots’ space: the next (but not final) frontier Getting from point A to point B robot navigation via path planning full-knowledge techniques and insect-inspired algorithms

Sources • • • Prof. Maja Mataric Dr. Fred Martin Bryce Tucker and former PSU students A. Ferworn, Prof. Gaurav Sukhatme, USC Robotics Research Laboratory Paul Hannah • Reuven Granot, Technion • Dodds, Harvey Mudd College