DOCSLIB.ORG

Kinematics of Robots

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Kinematics of Robots Kinematics of Robots Alba Perez Gracia c Draft date September 9, 2011 24 Chapter 2 Robot Kinematics Using Matrix Algebra 2.1 Overview In Chapter 1 we characterized the motion using vector and matrix algebra. In this chapter we introduce the application of 4 × 4 homogeneous transforms in traditional robot analysis, where the motion of a robot is described as a 4 × 4 homogoeneous matrix representing the position of the end-effector. This matrix is constructed as the composition of a series of local displacements along the chain. In this chapter we will learn how to calculate the forward and inverse kinematics of serial robots by using the Denavit-Hartenberg parameters to create the 4 × 4 homogeneous matrices. 2.2 Robot Kinematics From a kinematics point of view, a robot can be defined as a mechanical system that can be programmed to perform a number of tasks involving movement under automatic control. The main characteristic of a robot is its capability of movement in a six-dimensional space that includes translational and rotational coordinates. We model any mechanical system, including robots, as a series of rigid links connected by joints. The joints restrict the relative movement of adjacent links, and are generally powered and equipped with systems to sense this movement. 25 26 CHAPTER 2. ROBOT KINEMATICS USING MATRIX ALGEBRA The degrees of freedom of the robot, also called mobility, are defined as the number of independent parameters needed to specify the positions of all members of the system relative to a base frame. The most commonly used criteria to find the mobility of mechanisms is the Kutzbach-Gruebler formula. For a robot with n links (counting the base) and j joints, in which each joint i allows fi degrees of freedom, we compute the mobility as j X M = 6(n − 1) − (6 − fi): (2.1) i=1 It is important to remark that this formula does not give the correct answer for several types of closed, or parallel, robots. For a serial robot, the degrees of freedom are equal to the number of joints multiplied by the mobility allowed by each joint. The mobility indicates how many of the joints of the robot need to be actuated, also called active joints. In a serial robot, all joints are active. Once the mobility is defined and the active joints are identified, the two main problems of kinematic analysis of robots are the direct kinematics and the inverse kinematics. In the direct kinematics, also called forward kinematics, we define the position of the end-effector as a matrix which is a function of the angles or slides at each joint. If the joint variables are known, the position of the end-effector is completely specified. The inverse kinematics problem consists of finding the joint angles to reach a given position. For serial robots, the inverse kinematics problem is harder than the forward kinematics, while it is the opposite for parallel robots. The solutions of the direct kinematics define the workspace of the robot. This can be defined as the set of all possible positions of the end-effector, constructed using all possible values of the joint variables within their range. Remember that position means location plus orientation; the workspace of the robot is a six-dimensional subset of the six-dimensional space of rotations and translations. Because of the difficulty in visualizing the workspace, several subspaces have been defined [13]. • Reachable workspace (WR): Set of all locations of the origin of the end-effector frame that the robot can reach. It is a three-dimensional subset of the workspace. • Dextrous workspace (WD): Set of all locations of the origin of the end-effector frame that the robot can reach with any orientation. It is useful because the robot has full dexterity in this subspace, which ensures that any task can be performed within it. • Workspace with constant orientation (Wθ): Set of all locations of the origin of the end-effector frame that the robot can reach with a specified orientation for the end-effector. 2.3. BASIC JOINTS 27 Notice that [ \ Wθ = WR ; Wθ = WD (2.2) θ2range θ2range Figure 2.1 shows the typical sketch of the reachable workspace for a commercial robot. Figure 2.1: Reachable workspace, Adept Viper m650 robot (Adept Technologies, Inc.) 2.3 Basic joints Releaux [19] identified six basic types of joints that continuously share a surface of contact, that he called lower pairs. His classification is still used today in most robotics and mechanisms books. He defined revolute (R), cylindric (C), prismatic (P), spherical (S), helical (H) and plane (E) joints. In our research, we only consider the basic types of revolute and prismatic joints; any of the others can be obtained by combining the two basic types with certain additional constraints. In addition to these, we refer to the universal joint (T), which can be modeled as two revolute joints at right angles. See Figure 2.2. The revolute joint is a one-degree-of-freedom joint that allows a rotation of angle θ about an arbitrary located joint axis G = g + g0. The prismatic joint is also a one-degree-of-freedom joint, which allows translation along the direction h of the axis H. It is interesting to notice that the location of the prismatic axis does not matter for the movement of the prismatic joint. The cylindric joint is a general screw motion, in which the rotation about and the translation along the screw axis S are nonzero and independent. It can be constructed as the composition of a revolute joint 28 CHAPTER 2. ROBOT KINEMATICS USING MATRIX ALGEBRA Figure 2.2: Common types of joints and a prismatic joint with same direction. In a similar fashion, the universal (T) joint and the spherical (S) joint can be constructed from individual rotations. 2.4 Forward Kinematics Equations Following the traditional matrix kinematics equations, we will represent local transformations from joint to joint using 4 × 4 homogeneous matrices. The composition of these displacements will give as a result the displacement from the base to the end-effector. Angles will be measured usually taking the "extended" position as the origin. However, this is an arbitrary choice and any reference configuration can be used as the zero for the joint variables. In order to create the local transformations along the chain, we identify the lines locating the joint axes, Si, and we compute the common normal lines Ai;i+1 between joint Si and joint Si+1. We consider that the link joining two consecutive joints extends along the common normal line, regardless of its real shape. We assign local coordinate frames with the Z axis along the joint axis and the X axis along the com- mon normal line. Local displacements will take us from local coordinate frame to local coordinate frame, from the base to the end-effector, see Figure 2.3. 2.4.1 Denavit-Hartenberg parameters In 1955, Denavit and Hartenberg introduced the methodology of 4 × 4 homogeneous matrix trans- formations to analyze robotic systems, which became a standard tool in robotics. They shaped the local transformations along the chain as screw displacements about the Z and X axes, de- pendent on only four parameters. With the pass of the years, slightly different versions of the Denavit-Hartenberg parameters have been developed. Our notation is similar to Craig [11]. 2.4. FORWARD KINEMATICS EQUATIONS 29 Figure 2.3: Local transformations along the links of a robot. We locate the first local frame with the Z axis along the first joint axis S1 and the X axis along the common normal A12; the origin of the frame is located at the intersection of these two lines. Assume that a general displacement [G] locates this first local frame with respect to the fixed frame, as shown in Figure 2.3. The second local frame has the same X axis but now the Z axis is aligned with S2, with its origin at the intersection point of these two lines, while the third local frame has same Z axis as the second and the line A23 as its X axis. From the local frame attached to the last joint axis to the moving frame attached to the gripper we define a general displacement [H]. Only four parameters are needed to transform from local frame to local frame: two correspond to the transformation along the link, which is an X-screw displacement, and two more to perform a Z-screw displacement about the joint. These parameters are: • Twist angle αi−1;i: Angle between joint axes Si−1 and Si measured about the common normal line Ai−1;i. • Link length ai−1;i: Distance between joint axes Si−1 and Si measured along the common normal line Ai−1;i. • Joint angle θi: Angle between previous common normal line Ai−1;i and next common normal line Ai;i+1, measured about joint axes Si. Notice that if joint Si is a revolute joint, the joint angle will include the variable value of the joint rotation. 30 CHAPTER 2. ROBOT KINEMATICS USING MATRIX ALGEBRA Joint αi−1;i ai−1;i di θi 1 0 0 d1 θ1 2 α12 a12 d2 θ2 3 α23 a23 d3 θ3 Table 2.1: Denavit-Hartenberg parameters. • Offset di: Distance between previous common normal line Ai−1;i and next common normal line Ai;i+1, measured along joint axes Si. Notice that if joint Si is a prismatic joint, the offset will include the variable value of the slide. The Denavit-Hartenberg parameters describe in a simple way the geometry of the robot.
Load more

Recommended publications
  • Mobile Robot Kinematics
    Mobile Robot Kinematics We're going to start talking about our mobile robots now. There robots differ from our arms in 2 ways: They have sensors, and they can move themselves around. Because their movement is so different from the arms, we will need to talk about a new style of kinematics: Differential Drive. 1. Differential Drive is how many mobile wheeled robots locomote. 2. Differential Drive robot typically have two powered wheels, one on each side of the robot. Sometimes there are other passive wheels that keep the robot from tipping over. 3. When both wheels turn at the same speed in the same direction, the robot moves straight in that direction. 4. When one wheel turns faster than the other, the robot turns in an arc toward the slower wheel. 5. When the wheels turn in opposite directions, the robot turns in place. 6. We can formally describe the robot behavior as follows: (a) If the robot is moving in a curve, there is a center of that curve at that moment, known as the Instantaneous Center of Curvature (or ICC). We talk about the instantaneous center, because we'll analyze this at each instant- the curve may, and probably will, change in the next moment. (b) If r is the radius of the curve (measured to the middle of the robot) and l is the distance between the wheels, then the rate of rotation (!) around the ICC is related to the velocity of the wheels by: l !(r + ) = v 2 r l !(r − ) = v 2 l Why? The angular velocity is defined as the positional velocity divided by the radius: dθ V = dt r 1 This should make some intuitive sense: the farther you are from the center of rotation, the faster you need to move to get the same angular velocity.
    [Show full text]
  • Abbreviations and Glossary
    Appendix A Abbreviations and Glossary Abbreviations are defined and the mathematical symbols and notations used in this book are specified. Furthermore, the random number generator used in this book is referenced. A.1 Abbreviations arccos arccosine BiRRT Bidirectional rapidly growing random tree C-space Configuration space DH Denavit-Hartenberg DLR German aerospace center DOF Degree of freedom FFT Fast fourier transformation IK Inverse kinematics HRI Human-Robot interface LWR Light weight robot MMI Institute of Man-Machine interaction PCA Principal component analysis PRM Probabilistic road map RRT Rapidly growing random tree rulaCapMap Rula-restricted capability map RULA Rapid upper limb assessment SFE Shape fit error TCP Tool center point OV workspace overlap 130 A Abbreviations and Glossary A.2 Mathematical Symbols C configuration space K(q) direct kinematics H set of all homogeneous matrices WR reachable workspace WD dexterous workspace WV versatile workspace F(R,x) function that maps to a homogeneous matrix VRobot voxel space for the robot arm VHuman voxel space for the human arm P set of points on the sphere Np set of point indices for the points on the sphere No set of orientation indices OS set of all homogeneous frames distributed on a sphere MS capability map A.3 Mathematical Notations a scalar value a vector aT vector transposed A matrix AT matrix transposed < a,b > inner product 3 SO(3) group of rotation matrices ∈ IR SO(3) := R ∈ IR 3×3| RRT = I,detR =+1 SE(3) IR 3 × SO(3) A TB reference frame B given in coordinates of reference frame A a ceiling function a floor function A.4 Random Sampling In this book, the drawing of random samples is often used.
    [Show full text]
  • KINEMATIC ANALYSIS of a THUMB-EXOSKELETON SYSTEM for POST STROKE REHABILITATION By
    KINEMATIC ANALYSIS OF A THUMB-EXOSKELETON SYSTEM FOR POST STROKE REHABILITATION By Vikash Gupta Thesis Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mechanical Engineering August, 2010 Nashville, Tennessee Approved: Professor Nilanjan Sarkar Professor Derek Kamper Professor Robert Webster III ACKNOWLEDGEMENTS I will like to extend my sincere thanks to Dr. Nilanjan Sarkar for his continuous support and guidance, for his encouragement and advice throughout my studies. He has always been a source of inspiration for me. I would also like to thank Dr. Derek Kamper from Rehabilitation Institute of Chicago, for providing valuable comments on the thesis and being supportive throughout the research. My deep gratitude and sincere thanks goes to my parents and my sister, Swati Gupta. Their invaluable support, understanding and encouragement from far home has always helped me to continue my research. I would also like to thank Hari Kr. Voruganti, Mukul Singhee, Abhishek Chowdhury, Sayantan Chatterjee and Raktim Mitra for their support. My sincere thanks goes to Nino Dzotsenidze, who has always supported and believed in me and helped me make the right decisions at all times. Without her support and encouragement, it would have not been possible. I would also like to thank and express my sincere gratitude to my colleagues and friends Milind Shashtri, Furui Wang, Yu Tian, Jadav Das and Uttama Lahiri at Robotics and Autonomous Systems Laboratory for their support. Last but not the least, I want to express my humble thanks and appreciation to Suzanne Weiss from the Mechanical Engineering department for always being supportive.
    [Show full text]
  • Randomized Path Planning for Redundant Manipulators Without Inverse Kinematics
    Randomized Path Planning for Redundant Manipulators without Inverse Kinematics Mike Vande Weghe Dave Ferguson, Siddhartha S. Srinivasa Institute for Complex Engineered Systems Intel Research Pittsburgh Carnegie Mellon University 4720 Forbes Avenue Pittsburgh, PA 15213 Pittsburgh, PA 15213 [email protected] {dave.ferguson, siddhartha.srinivasa}@intel.com Abstract— We present a sampling-based path planning al- gorithm capable of efficiently generating solutions for high- dimensional manipulation problems involving challenging inverse kinematics and complex obstacles. Our algorithm extends the Rapidly-exploring Random Tree (RRT) algorithm to cope with goals that are specified in a subspace of the manipulator configuration space through which the search tree is being grown. Underspecified goals occur naturally in arm planning, where the final end effector position is crucial but the configuration of the rest of the arm is not. To achieve this, the algorithm bootstraps an optimal local controller based on the transpose of the Jacobian to a global RRT search. The resulting approach, known as Jacobian Transpose-directed Rapidly Exploring Random Trees (JT-RRTs), is able to combine the configuration space exploration of RRTs with a workspace goal bias to produce direct paths through complex environments extremely efficiently, without the need for any inverse kinematics. We compare our algorithm to a recently- developed competing approach and provide results from both simulation and a 7 degree-of-freedom robotic arm. I. INTRODUCTION Path planning for robotic systems operating in real envi- ronments is hard. Not only must such systems deal with the standard planning challenges of potentially high-dimensional and complex search spaces, but they must also cope with im- perfect information regarding their surroundings and perhaps Fig.
    [Show full text]
  • Nyku: a Social Robot for Children with Autism Spectrum Disorders
    University of Denver Digital Commons @ DU Electronic Theses and Dissertations Graduate Studies 2020 Nyku: A Social Robot for Children With Autism Spectrum Disorders Dan Stephan Stoianovici University of Denver Follow this and additional works at: https://digitalcommons.du.edu/etd Part of the Disability Studies Commons, Electrical and Computer Engineering Commons, and the Robotics Commons Recommended Citation Stoianovici, Dan Stephan, "Nyku: A Social Robot for Children With Autism Spectrum Disorders" (2020). Electronic Theses and Dissertations. 1843. https://digitalcommons.du.edu/etd/1843 This Thesis is brought to you for free and open access by the Graduate Studies at Digital Commons @ DU. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons @ DU. For more information, please contact [email protected],[email protected]. Nyku : A Social Robot for Children with Autism Spectrum Disorders A Thesis Presented to the Faculty of the Daniel Felix Ritchie School of Engineering and Computer Science University of Denver In Partial Fulfillment of the Requirements for the Degree Master of Science by Dan Stoianovici August 2020 Advisor: Dr. Mohammad H. Mahoor c Copyright by Dan Stoianovici 2020 All Rights Reserved Author: Dan Stoianovici Title: Nyku: A Social Robot for Children with Autism Spectrum Disorders Advisor: Dr. Mohammad H. Mahoor Degree Date: August 2020 Abstract The continued growth of Autism Spectrum Disorders (ASD) around the world has spurred a growth in new therapeutic methods to increase the positive outcomes of an ASD diagnosis. It has been agreed that the early detection and intervention of ASD disorders leads to greatly increased positive outcomes for individuals living with the disorders.
    [Show full text]
  • A Review of Parallel Processing Approaches to Robot Kinematics and Jacobian
    Technical Report 10/97, University of Karlsruhe, Computer Science Department, ISSN 1432-7864 A Review of Parallel Processing Approaches to Robot Kinematics and Jacobian Dominik HENRICH, Joachim KARL und Heinz WÖRN Institute for Real-Time Computer Systems and Robotics University of Karlsruhe, D-76128 Karlsruhe, Germany e-mail: [email protected] Abstract Due to continuously increasing demands in the area of advanced robot control, it became necessary to speed up the computation. One way to reduce the computation time is to distribute the computation onto several processing units. In this survey we present different approaches to parallel computation of robot kinematics and Jacobian. Thereby, we discuss both the forward and the reverse problem. We introduce a classification scheme and classify the references by this scheme. Keywords: parallel processing, Jacobian, robot kinematics, robot control. 1 Introduction Due to continuously increasing demands in the area of advanced robot control, it became necessary to speed up the computation. Since it should be possible to control the motion of a robot manipulator in real-time, it is necessary to reduce the computation time to less than the cycle rate of the control loop. One way to reduce the computation time is to distribute the computation over several processing units. There are other overviews and reviews on parallel processing approaches to robotic problems. Earlier overviews include [Lee89] and [Graham89]. Lee takes a closer look at parallel approaches in [Lee91]. He tries to find common features in the different problems of kinematics, dynamics and Jacobian computation. The latest summary is from Zomaya et al. [Zomaya96].
    [Show full text]
  • Arxiv:2102.12942V3 [Cs.RO] 30 Mar 2021 Drawback of These Strategies Is That They Are Extremely Sensitive to Inaccuracies in the Model
    Structured Prediction for CRiSP Inverse Kinematics Learning with Misspecified Robot Models Gian Maria Marconi∗,1,2 Raffaello Camoriano∗,2 Lorenzo Rosasco2,3,4 Carlo Ciliberto5 [email protected] raff[email protected] [email protected] [email protected] Abstract With the recent advances in machine learning, problems that traditionally would require accurate modeling to be solved analytically can now be successfully approached with data-driven strategies. Among these, computing the inverse kinematics of a redundant robot arm poses a significant challenge due to the non-linear structure of the robot, the hard joint constraints and the non-invertible kinematics map. Moreover, most learning algorithms consider a completely data-driven approach, while often useful information on the structure of the robot is available and should be positively exploited. In this work, we present a simple, yet effective, approach for learning the inverse kinematics. We introduce a structured prediction algorithm that combines a data-driven strategy with the model provided by a forward kinematics function – even when this function is misspecified – to accurately solve the problem. The proposed approach ensures that predicted joint configurations are well within the robot’s constraints. We also provide statistical guarantees on the generalization properties of our estimator as well as an empirical evaluation of its performance on trajectory reconstruction tasks. 1 Introduction Computing the inverse kinematics of a robot is a well-known key problem in several applications requiring robot control [20]. This task consists in finding a set of joint configurations that would result in a given pose of the end effector, and is traditionally solved by assuming access to an accurate model of the robot and employing geometric or numerical optimization techniques.
    [Show full text]
  • A Closed-Form Solution for the Inverse Kinematics of the 2N-DOF Hyper-Redundant Manipulator Based on General Spherical Joint
    applied sciences Article A Closed-Form Solution for the Inverse Kinematics of the 2n-DOF Hyper-Redundant Manipulator Based on General Spherical Joint Ya’nan Lou , Pengkun Quan, Haoyu Lin, Dongbo Wei and Shichun Di * School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China; [email protected] (Y.L.); [email protected] (P.Q.); [email protected] (H.L.); [email protected] (D.W.) * Correspondence: [email protected]; Tel.: +86-1390-4605-946 Abstract: This paper presents a closed-form inverse kinematics solution for the 2n-degree of freedom (DOF) hyper-redundant serial manipulator with n identical universal joints (UJs). The proposed algorithm is based on a novel concept named as general spherical joint (GSJ). In this work, these universal joints are modeled as general spherical joints through introducing a virtual revolution between two adjacent universal joints. This virtual revolution acts as the third revolute DOF of the general spherical joint. Remarkably, the proposed general spherical joint can also realize the decoupling of position and orientation just as the spherical wrist. Further, based on this, the universal joint angles can be solved if all of the positions of the general spherical joints are known. The position of a general spherical joint can be determined by using three distances between this unknown general spherical joint and another three known ones. Finally, a closed-form solution for the whole manipulator is solved by applying the inverse kinematics of single general spherical joint section using these positions. Simulations are developed to verify the validity of the proposed closed-form inverse kinematics model.
    [Show full text]
  • Acknowledgements Acknowl
    2161 Acknowledgements Acknowl. B.21 Actuators for Soft Robotics F.58 Robotics in Hazardous Applications by Alin Albu-Schäffer, Antonio Bicchi by James Trevelyan, William Hamel, The authors of this chapter have used liberally of Sung-Chul Kang work done by a group of collaborators involved James Trevelyan acknowledges Surya Singh for de- in the EU projects PHRIENDS, VIACTORS, and tailed suggestions on the original draft, and would also SAPHARI. We want to particularly thank Etienne Bur- like to thank the many unnamed mine clearance experts det, Federico Carpi, Manuel Catalano, Manolo Gara- who have provided guidance and comments over many bini, Giorgio Grioli, Sami Haddadin, Dominic Lacatos, years, as well as Prof. S. Hirose, Scanjack, Way In- Can zparpucu, Florian Petit, Joshua Schultz, Nikos dustry, Japan Atomic Energy Agency, and Total Marine Tsagarakis, Bram Vanderborght, and Sebastian Wolf for Systems for providing photographs. their substantial contributions to this chapter and the William R. Hamel would like to acknowledge work behind it. the US Department of Energy’s Robotics Crosscut- ting Program and all of his colleagues at the na- C.29 Inertial Sensing, GPS and Odometry tional laboratories and universities for many years by Gregory Dudek, Michael Jenkin of dealing with remote hazardous operations, and all We would like to thank Sarah Jenkin for her help with of his collaborators at the Field Robotics Center at the figures. Carnegie Mellon University, particularly James Os- born, who were pivotal in developing ideas for future D.36 Motion for Manipulation Tasks telerobots. by James Kuffner, Jing Xiao Sungchul Kang acknowledges Changhyun Cho, We acknowledge the contribution that the authors of the Woosub Lee, Dongsuk Ryu at KIST (Korean Institute first edition made to this chapter revision, particularly for Science and Technology), Korea for their provid- Sect.
    [Show full text]
  • Forward and Inverse Kinematics Analysis of Denso Robot
    Proceedings of the International Symposium of Mechanism and Machine Science, 2017 AzC IFToMM – Azerbaijan Technical University 11-14 September 2017, Baku, Azerbaijan Forward and Inverse Kinematics Analysis of Denso Robot Mehmet Erkan KÜTÜK 1*, Memik Taylan DAŞ2, Lale Canan DÜLGER1 1*Mechanical Engineering Department, University of Gaziantep Gaziantep/ Turkey E-mail: [email protected] 2 Mechanical Engineering Department, University of Kırıkkale Abstract used Robotic Toolbox in forward kinematics analysis of A forward and inverse kinematic analysis of 6 axis an industrial robot [4]. DENSO robot with closed form solution is performed in This study includes kinematics of robot arm which is this paper. Robotics toolbox provides a great simplicity to available Gaziantep University, Mechanical Engineering us dealing with kinematics of robots with the ready Department, Mechatronics Lab. Forward and Inverse functions on it. However, making calculations in kinematics analysis are performed. Robotics Toolbox is traditional way is important to dominate the kinematics also applied to model Denso robot system. A GUI is built which is one of the main topics of robotics. Robotic for practical use of robotic system. toolbox in Matlab® is used to model Denso robot system. GUI studies including Robotic Toolbox are given with 2. Robot Arm Kinematics simulation examples. Keywords: Robot Kinematics, Simulation, Denso The robot kinematics can be categorized into two Robot, Robotic Toolbox, GUI main parts; forward and inverse kinematics. Forward kinematics problem is not difficult to perform and there is no complexity in deriving the equations in contrast to the 1. Introduction inverse kinematics. Especially nonlinear equations make the inverse kinematics problem complex.
    [Show full text]
  • Kinematics, Kinematics Chains Previously
    Kinematics, Kinematics Chains Previously • Representation of rigid body motion • Two different interpretations - as transformations between different coordinate frames - as operators acting on a rigid body • Representation in terms of homogeneous coordinates • Composition of rigid body motions • Inverse of rigid body motion Rigid Body Transform Translation only is the origin of the frame B expressed in the Frame A Composite transformation: {B} Transformation: Homogeneous coordinates The points from frame A to frame B are {A} transformed by the inverse of (see example next slide) Kinematic Chains • We will focus on mobile robots (brief digression) • In general robotics - study of multiple rigid bodies lined together (e.g. robot manipulator) • Kinematics – study of position, orientation, velocity, acceleration regardless of the forces • Simple examples of kinematic model of robot manipulator and mobile robot • Components – links, connected by joints Various joints Kinematic Chains Tool frame Base frame • Given determine what is • Given determine what is • We can control , want to understand how it affects position of the tool frame • How does the position of the tool frame change as the manipulator articulates • Actuators change the joint angles Forward kinematics for a 2D arm • Find position of the end effector as a function of the joint angles • Blackboard example Kinematic Chains in 3D • More joints possible (spherical, screw) • Additional offset parameters, more complicated • Same idea: set up frame with each link • Define relationship
    [Show full text]
  • Lecture 10 Forward Kinematics Katie DC Feb
    lecture 10 forward kinematics Katie DC Feb. 25, 2020 Modern Robotics Ch. 4 Admin • Guest Lecture on Thursday 2/27 • Quiz 1 re-take is this week • HW3: PF Implementation due next week • HW5 due this week • I’ll be posting a course feedback form this week What is Forward Kinematics? • Kinematics: a branch of classical mechanics that describes motion of bodies without considering forces. AKA “the geometry of motion” • Forward kinematics: a specific problem in robotics. Given the individual state of each joint of the robot (in a local frame), what is the position of a given point on the robot in the global frame? Where is forward kinematics used? Forward Kinematics of a Simple Chain Assumptions • Our robot is a kinematic chain, made of rigid links and movable joints • No branches or loops (will discuss later) • All joints have one degree of freedom and are revolute or prismatic Review of Screw Motions We derived a representation for the screw axis: 휔 풮 = ∈ ℝ6 푣 where either • 휔 = 1 • where 푣 = −휔 × 푞 + ℎ휔, where 푞 is the point on the axis of the screw and ℎ is the pitch of the screw • 휔 = 0 and 푣 = 1 Screw Motions as Matrix Exponential The screw axis 풮푖 can be expressed in matrix form as: 휔 푣 풮 = 푖 ∈ 푠푒 3 푖 0 0 To express a screw motion given a screw axis, we use the matrix exponential: 푒 풮 휃 ∈ 푆퐸 3 Modeling Robot Joints as Screw Motions Case 1: Revolute Joint • 휔 = 1 • 푣 = −휔 × 푞 + ℎω • 푣 = −휔 × 푞 Modeling Robot Joints as Screw Motions Case 2: Prismatic Joint • 휔 = 0 • 푣 = 1 • Axis of movement defines 푣 Product of Exponentials Approach Let
    [Show full text]