Kinematics: Intro. Kinematics is study of motion

Transcription

1 Kinematics is study of motion Kinematics: Intro Concerned with mechanisms and how they transfer and transform motion Mechanisms can be machines, skeletons, etc. Important for CG since need to animate complex objects that may have many components or may be interconnected Motion of rigid objects involves Speed, velocity, and acceleration Degrees of freedom, constraints, and how they affect motion Two types of kinematics: 1. Forward 2. Inverse All of these important for computer animation as provide foundation on which all action is built 1

2 Link is basic kinematic component Link is rigid moving part Kinematics: Basic Concepts Linkage is set of links combined via joints Joint is movable connection Two types: 1. Pivot is based on rotation 2. Slide (piston) is based on translation Kinematic pair is simplest linkage: 2 links combined via joint 1 link constrained to pivot (or slide) Is primary component of kinematics Kinematic chain (articulated chain) is set of kinematic pairs linked by joints Proximal end is link of chain that is fixed to some anchor Root is point of articulation of proximal end Distal end is end of final link End effector is point of distal end that can be used to move the chain 2

3 Kinematics: Components (2) Closed kinematic chain is closed mechanical connection of kinematic pairs Proximal end connected to distal end At least 1 link is immobile 4 bar linkage is simplest closed kinematic chain Consists of 4 links and 4 joints Fixed link is immobile Driver link drives the motion Connected to one end of fixed link Follower link moved by driver link Connected to other end of fixed link Coupler link connects driver to follower Constrained so that given position of 1 link, others known Skeleton (armature) is hierarchy of kinematic chains Children can be transformed independently Each level has own coord system Objects rotate about own centers Branches rotate about parents centers Rotation of parent propagated to children 3

4 Kinematics: Models Kinematic model is kinematic chain and geometry that surrounds chain 1. Rigid geometry does not deform 2. Flexible geometry deforms Called skin or envelope Simple machine alters magnitude and/or direction of applied force 1. Lever 2. Pulley 3. Inclined plane 4. Wedge 5. Screw Machine (mechanism) is system of connected (usually) rigid bodies that alter, transmit, and direct force in predetermined way Combination of simple machines Machine parts characterized by their motion Translatory motion follows straight path Rotary motion follows curve or circular path Motion can be 1. Continuous and omnidirectional 2. Reciprocal and oscillatory Devices for transforming motion: Rocker arm reverses motion Bell crank amplifies motion Crank transmits rotary motion 4

5 Kinematics: Models - 4 Bar Linkages 8 basic 4 bar linkages associated with cranks Characterized by 1. Number of cranks 2. Conversion of motion 3. Velocity ratio of driver and follower NOTE: Driver on left Linkage Number of Motion converted Velocity cranks ratio Double lever 0 Reciprocating arc to Variable reciprocating arc Crank lever 1 Continuous circular to Variable (crank rocker) reciprocating arc Double crank 2 Continuous circular to Variable (drag link) continuous circular Parallel double 2 Continuous circular to Constant crank continuous circular equal velocity Reverse anti- 2 Continuous circular to Constant parallel reverse continuous circular opposite velocity Converse anti- 2 Continuous circular to Constant parallel reverse continuous circular equal velocity Isosceles double 2 Continuous circular to Constant crank continuous circular variable velocity Isosceles crank 1 Continuous circular to Variable lever reciprocating arc 5

6 Kinematics: Models - Motion Conversion Devices for converting one type of motion to another: To Continuous Reciprocating From Trans Rotary Trans Rotary arc Cont Rotary Double crank Cam Crank lever Parallel crank Slider crank Recip Trans Slider crank Rocker arm Slider lever (toggle) Rotary arc Lever crank Lever slider Double lever Slider-crank Connecting rod converts reciprocal to rotary motion Cam Lifts rod up and down 2 Types: 1. Continuous motion that is discontinuous at extremes 2. Continuous acceleration Devices can be combined in infinite ways Effectors can made to move freely in 3D 6

7 Kinematics: Analysis and Synthesis Kinematic synthesis is creation of geometric model or mechanical assembly that can trace a particular motion pathway Kinematic analysis is computation of a parameter (location, velocity, etc.) associated with any point or link of a mechanism in terms of the mechanism s components and their parameters Mechanisms can be classified by 1. Positional analysis Based on how mechanism moves 2. Contact analysis Based on type of contact between driver and follower: (a) Wrapping contact (b) Direct (rolling) contact (c) Rigid contact All have (a) Fixed link (b) Driver and follower, either rotating or reciprocating (c) Connector 3. Velocity analysis Classification based on (a) Speed of driver and follower (b) Ratio of speeds Contact type Speeds Ratios Wrapping May vary Constant Direct May vary May vary Rigid May vary May vary Wide variety of possibilities, e.g., (a) Velocity of follower oscillation Toggle position is extreme point at which v = 0 and direction changes (b) Time of travel to point differs from return time Quick-return device 7

8 Kinematics: Degrees of Freedom Degrees of freedom are number of ways a linkage can move independently (or number of parameters needed to describe orientation) Gruebler s equation determines degrees of freedom of mechanism Let total number of links NumLinks = N2 + N3 +...Nj, where Ni = number of links with i joints Total number of joints Joints = (2 N2 + 3 N J Nj)/2 Then DoF = 3 (NumLinks 1) 2 Joints 8

9 Kinematics: Forward and Inverse Kinematics (FK and IK) 2 approaches to animating a kinematic chain: Forward and Inverse FK: Start with root, specify angle of each link, moving from root toward leaves Allows full control of all joints Complex, requiring lots of work Animator specifies positions for each (key) frame Difficult to imitate natural motion Bottom-up approach IK: Specify position and orientation of end effector for start and end frames Software calculates joint angles required to go from start to final orientation Easy, as animator specifies goal only Produces set of possible solutions Top-down approach 9

10 Kinematics: IK Constraints Constraint is condition that imposes limitation on mechanism wrt position, orientation, velocity Can be positional and temporal Types of constraints: 1. Positional Constrains position of one object wrt others 2. Interpenetration Determines whether 2 objects can occupy the same space at the same time Dependent on object interaction - collision detection, contact analysis 3. Scaling Constrains size of one object wrt others 4. Rotational Constrains rotation of one object wrt others 5. Orientation Constrains orientation of one object in 3D space wrt others Constraint strength is relative importance of one constraint wrt others Fully constrained system is one in which the number constraints = degrees of freedom Results in predictable motion Over constrained system is one in which the number constraints > degrees of freedom Under constrained system is one in which the number constraints < degrees of freedom Degree of redundancy is DoF constraints Functional control system uses constraint strengths to find a solution when constraints conflict 10