Robotics

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Robotics

• Week 7
• Visualisation, Animation and VR

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Contents

• Robots
• Robot Arms
• Joints and linkages
• Inverse Kinematics
• Forces
• OpenGL solution

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What is a robot?

• Joseph Engelberger, a pioneer in industrial
  robotics, once remarked "I can't define a robot,
  but I know one when I see one."
• Many different machines called robots
• Everybody has a different idea of what
  constitutes a robot
• Name from robota forced labour

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What relevance to us?

• VR models use robotic principles
• Avatars behave like robots
• Simulations of robots used to test real robots
• May be used to control remote robotics

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Robotic Basics

• Have moveable segments
• Connected with joints
• Robots spin wheels and pivot jointed segments
  with some sort of actuator
• Some robots use electric motors and solenoids as
  actuators some use a hydraulic system and some
  use a pneumatic system (a system driven by
  compressed gases).
• Robots may use all these actuator types.
• Robots usually have some sort of sensor

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Actuators

• Electrical current drives actuators controlling
  individual joints
• Directly to motors or solenoids
• To valves controlling flow of fluids to hydraulic
  or pneumatic systems

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Robot arm

• Simplest sort of robot
• Typical arm has 7 segments, 6 joints
• 6DOF
• Human arm 7DOF
• Usually driven by Step Motors
• Main use is in manufacturing

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Robot Arm

• Fitted with end effector
• Usually interchangeable
• Artificial Hand , paint gun, welding rod
• Pressure sensor needed to prevent crushing
• Programmed by incremental steps which are then
  replicated ad infinitum

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Step Motor

• electromagnetic, rotary actuator, that
  mechanically converts digital pulse inputs to
  incremental shaft rotation.
• The rotation not only has a direct relation to
  the number of input pulses, but its speed is
  related to the frequency of the pulses.

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Step Motor
Each pulse corresponds to an angular rotation
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Step Motors

• Between steps holds position w/o brake or clutch
• Can be programmed to move a precise number of
  steps and then hold position
• Possible to be bi-directional
• Rapid acceleration, deceleration and reversal
• cf DC Servo motors

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Choosing the right motor

• Basic Types
• Variable Reluctance,
• Permanent Magnet,
• Hybrid
• Parameters to be considered
• Distance to be traversed.
• Maximum time allowed for a traverse.
• Desired detent (static) accuracy.
• Desired dynamic accuracy (overshoot).

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More parameters

• Settling time
• Required step resolutiong
• System friction
• System inertia.
• Speed/Torque characteristics of the motor When
  selecting a motor/drive, the capacity of the
  motor must exceed the overall requirements of the
  load.
• Torque-to-inertia Ratio
• Torque Margin Selecting a motor drive that
  provides at least 50 margin above the minimum
  required torque is ideal.

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Frameworks, Chains (or Skeletons)

• A lot of mechanical objects in the real world
  consist of solid sections connected by joints
• Obviously robot arm but also
• Creatures such as humans and animals.
• Car Suspension
• Ropes, string and Chains

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Frameworks, Chains (or Skeletons)

• Sections and joints of robot arm are known as a
  'chain
• In creatures could be referred to as a skeleton
• Moveable sections correspond to bones
• Attachments between bones are joints.

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Frameworks, Chains (or Skeletons)

• Motions of chains can be specified in terms of
  translations and rotations.
• Forward Kinematics - From the amounts of rotation
  and bending of each joint in an arm, for example,
  the position of the hand can be calculated.
• Inverse Kinematics - If the hand is moved, the
  rotation and bending of the arm is calculated, in
  accordance with the length and joint properties
  of each section of the arm.

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Joint Translation-Rotation

• We can use a transform (T) to transform each
  point relative to the body to a position in world
  coordinates.
• If we want to model both linear and angular
  (rotational) motion then we need to use a 4x4
  matrix to represent the transform

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What is Inverse Kinematics?

• Forward Kinematics

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What is Inverse Kinematics?

• Inverse Kinematics

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What does looks like?
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Solution to

• Our example

Number of equation 2
Unknown variables 3
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Redundancy

• System DOF gt End Effector DOF

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Redundancy

• A redundant system has infinite number of
  solutions
• Human skeleton has 70 DOF
• Ultra-super redundant
• How to solve highly redundant system?

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Iterative solution

• Start at end effector
• Move each joint so that end gets closer to target
• The angle of rotation for each joint is found by
  taking the dot product of the vectors from the
  joint to the current point and from the joint to
  the desired end point. Then taking the arcsin of
  this dot product.
• To find the sign of this angle (ie which
  direction to turn), take the cross product of
  these vectors and checking the sign of the Z
  element of the vector.

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Goal Potential Function

• Distance from the end effector to the goal
• Function of joint angles G(q)

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Our Example
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Quiz

• Will G(q) be always zero?
• No Unreachable Workspace
• Will the solution be always found?
• No Local Minima/Singular Configuration
• Will the solution be always unique?
• No Redundancy

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Conflict Between Goals
ee 2
ee 1
base
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Conflict Between Goals
Goal 1
ee 2
ee 1
base
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Conflict Between Goals
Goal 2
ee 2
ee 1
base
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Conflict Between Goals
Goal 2
Goal 1
ee 2
ee 1
base
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Conflict Between Goals
Goal 2
Goal 1
ee 2
ee 1
base
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Joint Structures

• This allows two nodes to be attached to each
  other in a flexible way so that forces in the
  plane of the joint will be transmitted through
  it, but forces perpendicular to the joint will
  cause it to bend. This will provide IK like
  capabilities

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Types of Joint


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Joint Structures

• In character animation, only 2 types of joint
  need to be considered. These are the "revolute"
  and "prismatic" joints. All other types can be
  based on these two.
• 1 degree of freedom
• rotational joint - wheel.
• hinge - similar to rotational joint above but
  with limits to motion (end stops)
• 2 degrees of freedom
• ball socket joint

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Dynamics

• Forward Dynamics - The movements are calculated
  from the forces, such as, force mass
  acceleration.
• Inverse Dynamics - Constraints are applied which
  specifies how objects interact, for example, they
  may be linked by a hinge joint or a ball joint,
  and from this the forces can be calculated

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Forward Dynamics

• If no forces act on a particle, the particle
  retains its linear momentum.
• The rate of change of the linear momentum of a
  particle is equal to the sum of all forces acting
  on it.
• When two particles exert forces upon each other,
  these forces are equal in magnitude and opposite
  in direction.

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Forward Dynamics

• These laws can also be applied to rigid bodies by
  assuming that the forces are acting on the centre
  of mass of the object.
• Assuming that the mass is constant then the
  second law becomes
• force mass acceleration
• Euler extended these laws to include rotation. So
  there are equivalent laws for rotation such as
• torque inertia angular acceleration.

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Steady State

• For object subject to no external forces
• The linear velocity of the centre-of-mass will be
  constant.
• The angular velocity about the centre-of-mass
  will be constant, and if the object is not
  symmetrical, this rotation will always be about
  one of its Principal Moments of Inertia.
• With linear motion (provided there is no air
  resistance) it can move in all directions
  equally, regardless of the shape of the object.
• With rotation the direction of spin may be
  dependent on the orientation of the object (and
  visa versa).

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Non-Steady State

• If a force is applied to the centre-of-mass then
  the object will accelerate (at a rate of F/m) and
  the angular velocity will not be affected.
• If a torque is applied about the centre-of-mass
  and along the axis of one of the principle
  moments of inertia then the angular speed will
  change (by T/I) and the linear velocity will not
  be affected.
• If force not thro centre will cause linear and
  angular variation

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Forces about a hinge

• Assume two masses, A and B, connected with a
  hinge. The only external forces acting on the
  system are
• Fa external force acting on the centre of mass
  of mass A.
• Fb external force acting on the centre of mass
  of mass B.

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Forces about a hinge

• And two torques acting on the system
• Ta external torque acting around the centre of
  mass of mass A.
• Tb external torque acting around the centre of
  mass of mass B.

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Forces about a hinge

• There are no other external forces acting on the
  system. A force may be transmitted between the
  two masses Fc
• By Newtons third law, the force of b of a is
  equal and opposite to the force of a on b. There
  is no torque between the two masses since a hinge
  is used.

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Forces about a hinge

• Therefore the dynamics equations are
• Fa Fc ma aa
• Fb - Fc mb ab
• Where
• Fa external force on mass 'a' (vector)
• Fb external force on mass 'b' (vector)
• Fc force transmitted between hinge (vector)
• ma mass of 'a' (scalar)
• aa linear acceleration of 'a' (vector)
• mb mass of 'b' (scalar)
• ab acceleration of 'b' (vector)

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Resultant Motion
Constrained motion
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Recap

• Robotic behaviour needs to be modelled
• Robot arm is a type of chain
• Movement of chain exhibits kinematics
• Motion of end effector derived from sum of
  movements of segments
• Can calculate the angular velocity and force of
  end point from combination of forces applied to
  individual joints in the assembly.