5. Introduction to Robot Geometry and Kinematics


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1 V. Kumar 5. Introduction to Robot Geometry and Kinematics The goa of this chapter is to introduce the basic terminoogy and notation used in robot geometry and kinematics, and to discuss the methods used for the anaysis and contro of robot manipuators. The scope of this discussion wi be imited, for the most part, to robots with panar geometry. The anaysis of manipuators with threedimensiona geometry can be found in any robotics text. 5. Some definitions and exampes We wi use the term mechanica system to describe a system or a coection of rigid or fexibe bodies that may be connected together by joints. A mechanism is a mechanica system that has the main purpose of transferring motion and/or forces from one or more sources to one or more outputs. A inkage is a mechanica system consisting of rigid bodies caed inks that are connected by either pin joints or siding joints. In this section, we wi consider mechanica systems consisting of rigid bodies, but we wi aso consider other types of joints. Degrees of freedom of a system The number of independent variabes (or coordinates) required to competey specify the configuration of the mechanica system. Whie the above definition of the number of degrees of freedom is motivated by the need to describe or anayze a mechanica system, it aso is very important for controing or driving a mechanica system. It is aso the number of independent inputs required to drive a the rigid bodies in the mechanica system. Exampes: (a) A point on a pane has two degrees of freedom. A point in space has three degrees of freedom. (b) A penduum restricted to swing in a pane has one degree of freedom. In particuar, two books offer an exceent treatment whie keeping the mathematics at a very simpe eve: (a) Craig, J. J. Introduction to Robotics, AddisonWesey, 989; and (b) Pau, R., Robot Manipuators, Mathematics, Programming and Contro, The MIT Press, Cambridge,
2 (c) A panar rigid body (or a amina) has three degrees of freedom. There are two if you consider transations and an additiona one when you incude rotations. (d) The mechanica system consisting of two panar rigid bodies connected by a pin joint has four degrees of freedom. Specifying the position and orientation of the first rigid body requires three variabes. Since the second one rotates reative to the first one, we need an additiona variabe to describe its motion. Thus, the tota number of independent variabes or the number of degrees of freedom is four. (e) A rigid body in three dimensions has six degrees of freedom. There are three transatory degrees of freedom. In addition, there are three different ways you can rotate a rigid body. For exampe, consider rotations about the x, y, and z axes. It turns out that any rigid body rotation can be accompished by successive rotations about the x, y, and z axes. If the three anges of rotation are considered to be the variabes that describe the rotation of the rigid body, it is evident there are three rotationa degrees of freedom. (f) Two rigid bodies in three dimensions connected by a pin joint have seven degrees of freedom. Specifying the position and orientation of the first rigid body requires six variabes. Since the second one rotates reative to the first one, we need an additiona variabe to describe its motion. Thus, the tota number of independent variabes or the number of degrees of freedom is seven. Kinematic chain A system of rigid bodies connected together by joints. A chain is caed cosed if it forms a cosed oop. A chain that is not cosed is caed an open chain. Seria chain If each ink of an open chain except the first and the ast ink is connected to two other inks it is caed a seria chain. An exampe of a seria chain can be seen in the schematic of the PUMA 560 series robot, an industria robot manufactured by Unimation Inc., shown in Figure. The trunk is boted to a fixed tabe or the foor. The shouder rotates about a vertica axis with respect to the trunk. The upper arm rotates about a horizonta axis with respect to the shouder. This rotation is the shouder joint rotation. The forearm rotates about a horizonta axis (the ebow) with respect to the upper arm. Finay, the wrist consists of an assemby of three rigid bodies with three The Programmabe Universa Machine for Assemby (PUMA) was deveoped in 978 by Unimation Inc. using a set of specifications provided by Genera Motors. Robot Geometry and Kinematics  V. Kumar
3 additiona rotations. Thus the robot arm consists of seven rigid bodies (the first one is fixed) and six joints connecting the rigid bodies. Figure The six degreeoffreedom PUMA 560 robot manipuator. Figure The six degreeoffreedom T robot manipuator. Robot Geometry and Kinematics  V. Kumar
4 Another schematic of an industria robot arm, the T made by Cincinnati Miacron, is shown in Figure. Once again, it is possibe to mode it as a coection of seven rigid bodies (the first being fixed) connected by six joints. Types of joints There are mainy four types of joints that are found in robot manipuators: Revoute, rotary or pin joint (R) Prismatic or siding joint (P) Spherica or ba joint (S) Heica or screw joint (H) The revoute joint aows a rotation between the two connecting inks. The best exampe of this is the hinge used to attach a door to the frame. The prismatic joint aows a pure transation between the two connecting inks. The connection between a piston and a cyinder in an interna combustion engine or a compressor is via a prismatic joint. The spherica joint between two inks aows the first ink to rotate in a possibe ways with respect to the second. The best exampe of this is seen in the human body. The shouder and hip joints, caed ba and socket joints, are spherica joints. The heica joint aows a heica motion between the two connecting bodies. A good exampe of this is the reative motion between a bot and a nut. Panar chain A the inks of a panar chain are constrained to move in or parae to the same pane. A panar chain can ony aow prismatic and revoute joints. In fact, the axes of the revoute joints must be perpendicuar to the pane of the chain whie the axes of the prismatic joints must be parae to or ie in the pane of the chain. An exampe of a panar chain is shown in Figure. Amost a interna combustion engines use a sider crank mechanism. The high pressure of the expanding gases in the combustion chamber is used to transate the piston and the mechanism converts this transatory movement into the rotary movement of the crank. This mechanica system consists of three revoute joints and one prismatic joint. The exampe in Figure is a panar, cosed, kinematic chain. Exampes of panar, seria chains are shown in Figure 4 and 5. This is a convenient mode. A more accurate kinematic mode is required to mode the couping between the actuator that drives the ebow joint and the ebow joint. Robot Geometry and Kinematics 4 V. Kumar
5 Connectivity of a joint The number of degrees of freedom of a rigid body connected to a fixed rigid body through the joint. The revoute, prismatic and heica joint have a connectivity. The spherica joint has a connectivity of. Sometimes one uses the term degree of freedom of a joint instead of the connectivity of a joint. Crank shaft τ r θ x Connecting rod φ Piston Figure A schematic of a sider crank mechanism Cyinder F ENDEFFECTOR R Link R Link Joint Joint Link R Joint ACTUATORS Figure 4 A schematic of a panar manipuator with three revoute joints Robot Geometry and Kinematics 5 V. Kumar
6 ENDEFFECTOR R P R ACTUATORS Figure 5 A schematic of a panar manipuator with two revoute and one prismatic joints Mobiity The mobiity of a chain is the number of degrees of freedom of the chain. Most books wi use the term number of degrees of freedom for the mobiity. In a seria chain, the mobiity of the chain is easiy cacuated. If there are n joints and joint i has a connectivity f i, M = n f i i= Most industria robots have either revoute or prismatic joints (f i = ) and therefore the mobiity or the number of degrees of freedom of the robot arm is aso equa to the number of joints. Sometimes, an n degree of freedom robot or a robot with mobiity n is aso caed an n axis robot. Since a rigid body in space has six degrees of freedom, the most genera robots are designed to have six joints. This way, the end effector or the ink that is furthest away from the base can be made to assume any position or orientation (within some range). However, if the end effector needs to moved around in a pane, the robot need ony have three degrees of freedom. Two exampes 4 of panar, three degree of freedom robots (technicay, mobiity three robots) are shown in Figures 4 and 5. 4 Note that we do not count the opening and cosing of the gripper as a degree of freedom. The gripper is usuay competey open or competey shut and it is not continuousy controed as the other joints are. Aso, the gripper freedom does not participate in the positioning and orienting of a part hed by the gripper. Robot Geometry and Kinematics 6 V. Kumar
7 When cosed oops are present in the kinematic chain (that is, the chain is no onger seria, or even open), it is more difficut to determine the number of degrees of freedom or the mobiity of the robot. But there is a simpe formua that one can derive for this purpose. Let n be the number of moving inks and et g be the number of joints, with f i being the connectivity of joint i. Each rigid body has six degrees if we consider spatia motions. If there were no joints, since there are n moving rigid bodies, the system woud have 6n degrees of freedom. The effect of each joint is to constrain the reative motion of the two connecting bodies. If the joint has a connectivity f i, it imposes (6f i ) constraints on the reative motion. In other words, since there are f i different ways for one body to move reative to another, there (6f i ) different ways in which the body is constrained from moving reative to another. Therefore, the number of degrees of freedom or the mobiity of a chain (incuding the specia case of a seria chain) is given by: M = 6 n g ( 6 f i ) i= or, M g f i i= ( n g) + = 6 () ENDEFFECTOR ACTUATORS Figure 6 A panar parae manipuator. Robot Geometry and Kinematics 7 V. Kumar
8 In the specia case of panar motion, since each unconstrained rigid body has degrees of freedom, this equation is modified to: g f i i= ( n g) + M = () Exampe In Figures 4 and Figure 5, since n=g=, Equation () reduces to the specia case of Equation (). And since f = f = f =, and M=g. Exampe In the sider crank mechanism shown in Figure, n= andg=4. Since it is a panar mechanism we use Equation (). A four joints have connectivity one: f = f = f = f 4 =, and M=. Exampe Consider the parae manipuator shown in Figure 6. Here, n = 7, g=9, and f i =. According to Equation (), M =. There are correspondingy three actuators in the manipuator. Contrast this arrangement with the arrangement shown in Figures 4 and 5. The three actuators are mounted in parae in Figure 6. In Figures 4 and 5, they are mounted sequentiay in a seria fashion. The Stewart Patform The StewartGough or the Stewart Patform 5 device is a six degree of freedom (mobiity six) kinematic chain with cosed oops. The kinematic chain consists of a base and a moving patform each of which is a spatia hexagon. See Figure 7. Every vertex of the base hexagon is connected to one vertex of the moving patform hexagon by one eg. Simiary, every vertex of the moving hexagon is connected to a vertex of the base hexagon by a eg. There are six such egs. Each eg has is a seria chain consisting of two revoute joints with intersecting axes, a prismatic joint and a spherica joint. Typicay the prismatic joints are actuated. The mobiity of a Stewart Patform can be easiy verified to be six. Each eg has three inks and four joints. If we incude the moving patform, n = 6 + = 9. 5 D. Stewart, A Patform with Six Degrees of Freedom, The Institution of Mechanica Engineers, Proceedings , Vo. 80 Part, No. 5, pages Robot Geometry and Kinematics 8 V. Kumar
9 (a) A machine too based on the Stewart Patform (Ingerso Rand) 6 S END EFFECTOR Leg 5 P Leg 4 Leg 6 Leg Leg Leg R R BASE LEG NO. i (b) A schematic showing the six egs (eft) and the RRPS chain (right). Figure 7 The Stewart Patform 6 M. Vaenti, Machine Toos Get Smarter, Mechanica Engineering, Vo.7, No., November 995. Robot Geometry and Kinematics 9 V. Kumar
10 The connectivity of the revoute and the prismatic joint is one. The connectivity of the spherica joint is three. Since there are 6 revoute joints, 6 prismatic joints and 6 spherica joints, g i= f i = = 6 According to Equation (), M = 69 ( 4)+ 6 = 6 The Stewart Patform has actuators for a its six prismatic joints and it is therefore possibe to contro a six degrees of freedom. (a) The Adept 850 Paetizer (b) side view (axes 4 are numbered) (c) top view (axes 4 are numbered) Figure 8 The Adept 850 Paetizer There are four degrees of freedom in this SCARA manipuator. Joint is a siding joint that carries the manipuator arm up or down. Joints 4 are rotary joints with vertica axes. Robot Geometry and Kinematics 0 V. Kumar
11 5. Geometry of panar robot manipuators The mathematica modeing of spatia inkages is quite invoved. It is usefu to start with panar robots because the kinematics of panar mechanisms is generay much simper to anayze. Aso, panar exampes iustrate the basic probems encountered in robot design, anaysis and contro without having to get too deepy invoved in the mathematics. However, whie the exampes we wi discuss wi invove kinematic chains that are panar, a the definitions and ideas presented in this section are genera and extend to the most genera spatia mechanisms. We wi start with the exampe of the panar manipuator with three revoute joints. The manipuator is caed a panar R manipuator. Whie there may not be any three degree of freedom (d.o.f.) industria robots with this geometry, the panar R geometry can be found in many robot manipuators. For exampe, the shouder swive, ebow extension, and pitch of the Cincinnati Miacron T robot (Figure ) can be described as a panar R chain. Simiary, in a four d.o.f. SCARA manipuator (Figure 8), if we ignore the prismatic joint for owering or raising the gripper, the other three joints form a panar R chain. Thus, it is instructive to study the panar R manipuator as an exampe. In order to specify the geometry of the panar R robot, we require three parameters,,, and. These are the three ink engths. In Figure 9, the three joint anges are abeed θ, θ, and θ. These are obviousy variabe. The precise definitions for the ink engths and joint anges are as foows. For each pair of adjacent axes we can define a common norma or the perpendicuar between the axes. The ith common norma is the perpendicuar between the axes for joint i and joint i+. The ith ink ength is the ength of the ith common norma, or the distance between the axes for joint i and joint i+. The ith joint ange is the ange between the (i)th common norma and ith common norma measured counter cockwise going from the (i)th common norma to the ith common norma. Note that there is some ambiguity as far as the ink ength of the most dista ink and the joint ange of the most proxima ink are concerned. We define the ink ength of the most dista ink from the most dista joint axis to a reference point or a too point on the end effector 7. Generay, this is the center of the gripper or the end point of the too. Since there is no zeroth common norma, we measure the first joint ange from a convenient reference ine. Here, we have chosen this to be the x axis of a convenienty defined fixed coordinate system. 7 The reference point is often caed the too center point (TCP). Robot Geometry and Kinematics  V. Kumar
12 Another set of variabes that is usefu to define is the set of coordinates for the end effector. These coordinates define the position and orientation of the end effector. With a convenient choice of a reference point on the end effector, we can describe the position of the end effector using the coordinates of the reference point (x, y) and the orientation using the ange φ. The three end effector coordinates (x, y, φ) competey specify the position and orientation of the end effector 8. REFERENCE POINT (x, y) φ θ y θ θ x Figure 9 The joint variabes and ink engths for a R panar manipuator As another exampe, consider the three d.o.f. cyindrica robot in Figure 0. If we ignore the ift freedom, the rotation of the base and the extension of the arm give us the two d.o.f. robot shown in Figure that we can ca the RP manipuator. It consists of a revoute joint and prismatic joint as shown in the figure. θ, the base rotation, and d, the arm extension, are the two joint variabes. Note that there are no constant parameters such as the three ink engths in the R manipuator. The joint variabe θ is defined as before. Since there is no zeroth common norma, 8 The description of the position and orientation of a three dimensiona rigid body is significanty more compicated. For a spatia manipuator, a typica set of end effector coordinates woud incude three variabes (x, y, z) for the position, and three Euer anges (θ, φ, ψ) for the orientation. Robot Geometry and Kinematics  V. Kumar
13 we measure the joint ange from the x axis which we have chosen to be horizonta. d is defined as the distance from joint axis to the reference point on the end effector. As in the previous exampe, the end effector coordinates are variabes that competey specify the position and orientation of the end effector. In the figure, they are (x, y, φ). Finay, we consider a Cartesian robot consisting of two prismatic joints at right anges. The PP chain is found in xy tabes, potters and miing machines. A schematic is shown in Figure. The simpest spatia manipuator is based on the PPP chain, which has a third prismatic joint. The three joint axes are mutuay orthogona. The Gantry robot in Figure has this geometry. If you ignore the vertica up/down degree of freedom it is a PP manipuator. Figure 0 The RT00 cyindrica robot (Seiko) REFERENCE POINT (x, y) y d φ θ x Figure The joint variabes and ink engths for a RP panar manipuator Robot Geometry and Kinematics  V. Kumar
14 d d Figure The joint variabes for a PP panar manipuator Figure The G65 Gantry robot manipuator (CRS Robotics) on the eft, and the Biomek 000 Laboratory Automation Workstation (Beckman Couter) on the right both have tooing mounted at the end of a PPP chain. The end effector of a manipuator that has ony prismatic joints is constrained to remain in the same orientation. Thus, the end effector coordinates for the PP manipuator ony incude the coordinates of the reference point on the end effector (x, y). In summary, in each case, we defined a set of constant parameters caed ink engths ( i ) and set of joint variabes or joint coordinates consisting of either joint anges (θ i ) or dispacements Robot Geometry and Kinematics 4 V. Kumar
15 (d i ). We aso defined a set of variabes caed end effector coordinates. The ink engths are constant parameters that define the geometry of the manipuator. The joint variabes define the configuration of the manipuator by specifying the position of each joint. The end effector coordinates define the position and orientation of the end effector. If the joint coordinates specify the configuration of the manipuator, they shoud aso specify the position and orientation of the end effector. Thus one shoud expect to find an expicit dependence of the end effector coordinates on the joint coordinates. Athough it may not be obvious, there is aso a dependence of the joint coordinates on the end effector coordinates. The next subsection wi address this dependence and anayse the kinematics of robot manipuators. 5. Kinematic anaysis of panar seria chains Kinematics is the study of motion. In this subsection, we wi expore the reationship between joint movements and end effector movements. More precisey, we wi try to deveop equations that wi make expicit the dependence of end effector coordinates on joint coordinates and vice versa. We wi start with the exampe of the panar R manipuator. From basic trigonometry, the position and orientation of the end effector can be written in terms of the joint coordinates in the foowing way: x = y = φ = cosθ + cos( θ + θ ) + cos( θ + θ + θ) sin θ + sin( θ + θ ) + sin( θ + θ + θ) ( θ + θ + θ ) Note that a the anges have been measured counter cockwise and the ink engths are assumed to be positive going from one joint axis to the immediatey dista joint axis. Equation (4) is a set of three noninear equations 9 that describe the reationship between end effector coordinates and joint coordinates. Notice that we have expicit equations for the end effector coordinates in terms of joint coordinates. However, to find the joint coordinates for a given set of end effector coordinates (x, y, φ), one needs to sove the noninear equations for θ, θ, and θ. The kinematics of the panar RP manipuator is easier to formuate. The equations are: (4) 9 The third equation is inear but the system of equations is noninear. Robot Geometry and Kinematics 5 V. Kumar
16 x = d y = d φ = θ cos θ sin θ Again the end effector coordinates are expicity given in terms of the joint coordinates. However, since the equations are simper (than in (4)), you woud expect the agebra invoved in soving for the joint coordinates in terms of the end effector coordinates to be easier. Notice that in contrast to (4), now there are three equations in ony two joint coordinates, θ, and d. Thus, in genera, we cannot sove for the joint coordinates for an arbitrary set of end effector coordinates. Said another way, the robot cannot, by moving its two joints, reach an arbitrary end effector position and orientation. Let us instead consider ony the position of the end effector described by (x, y), the coordinates of the end effector too point or reference point. We have ony two equations: (5) x = d y = d cos θ sin θ Given the end effector coordinates (x, y), the joint variabes can be computed to be: (6) d θ = + = tan x + y y x Notice that we restricted d to positive vaues. A negative d may be physicay achieved by aowing the end effector reference point to pass through the origin of the xy coordinate system over to another quadrant. In this case, we obtain another soution: (7) d θ = = tan x + y y x In both cases (78), the inverse tangent function is mutivaued 0. In particuar, (8) tan(x) = tan(x + kπ), k= , , 0,,, (9) However, if we imit θ to the range 0<θ <π, there is a unique vaue of θ that is consistent with the given (x, y) and the computed d (for which there are two choices). 0 In Appendix, we define another inverse tangent function caed atan that takes two arguments, the sine and the cosine of an ange, and returns a unique ange in the range [0, π). Robot Geometry and Kinematics 6 V. Kumar
17 The existence of mutipe soutions is typica when we sove noninear equations. As we wi see ater, this poses some interesting questions when we consider the contro of robot manipuators. The panar Cartesian manipuator is trivia to anayze. The equations for kinematic anaysis are: x = d, y = d (0) The simpicity of the kinematic equations makes the conversion from joint to end effector coordinates and back trivia. This is the reason why PP chains are so popuar in such automation equipment as robots, overhead cranes, and miing machines. Direct kinematics As seen earier, there are two types of coordinates that are usefu for describing the configuration of the system. If we focus our attention on the task and the end effector, we woud prefer to use Cartesian coordinates or end effector coordinates. The set of a such coordinates is generay referred to as the Cartesian space or end effector space. The other set of coordinates is the so caed joint coordinates that is usefu for describing the configuration of the mechanica nkage. The set of a such coordinates is generay caed the joint space. In robotics, it is often necessary to be abe to map joint coordinates to end effector coordinates. This map or the procedure used to obtain end effector coordinates from joint coordinates is caed direct kinematics. For exampe, for the R manipuator, the procedure reduces to simpy substituting the vaues for the joint anges in the equations x = y = φ = cosθ + cos( θ + θ ) + cos( θ + θ + θ) sin θ + sin( θ + θ ) + sin( θ + θ + θ) ( θ + θ + θ ) and determining the Cartesian coordinates, x, y, and φ. For the other exampes of open chains discussed so far (RP, PP) the process is even simper (since the equations are simper). In fact, for a seria chains (spatia chains incuded), the direct kinematics procedure is fairy straight forward. On the other hand, the same procedure becomes more compicated if the mechanism contains one or more cosed oops. In addition, the direct kinematics may yied more than one soution or Since each member of this set is an ntupe, we can think of it as a vector and the space is reay a vector space. But we sha not need this abstraction here. Robot Geometry and Kinematics 7 V. Kumar
18 no soution in such cases. For exampe, in the panar parae manipuator in Figure, the joint positions or coordinates are the engths of the three teescoping inks (q, q, q ) and the end effector coordinates (x, y, φ) are the position and orientation of the foating triange. It can be shown that depending on the vaue of (q, q, q ), the number of (rea) soutions for (x, y, φ) can be anywhere from zero to six. For the Stewart Patform in Figure 4, this number has been shown to be anywhere from zero to forty. 5.4 Inverse kinematics The anaysis or procedure that is used to compute the joint coordinates for a given set of end effector coordinates is caed inverse kinematics. Basicay, this procedure invoves soving a set of equations. However the equations are, in genera, noninear and compex, and therefore, the inverse kinematics anaysis can become quite invoved. Aso, as mentioned earier, even if it is possibe to sove the noninear equations, uniqueness is not guaranteed. There may not (and in genera, wi not) be a unique set of joint coordinates for the given end effector coordinates. We saw that for the RP manipuator, the direct kinematics equations are: x = d y = d cos θ sin θ (6) If we restrict the revoute joint to have a joint ange in the interva [0, π), there are two soutions for the inverse kinematics: d y = atan d x, d, = σ x + y, θ σ = ± Here we have used the atan function in Appendix to uniquey specify the joint ange θ. However, depending on the choice of σ, there are two soutions for d, and therefore for θ. The inverse kinematics anaysis for a panar R manipuator appears to be compicated but we can derive anaytica soutions. Reca that the direct kinematics equations (4) are: ( θ + θ ) + ( θ + θ + θ ) ( θ + θ ) + ( θ + θ + θ ) x = (4a) cosθ + cos cos sin θ + sin sin ( θ + θ + θ ) y = (4b) φ = (4c) The ony case in which the anaysis is trivia is the PP manipuator. In this case, there is a unique soution for the inverse kinematics. Robot Geometry and Kinematics 8 V. Kumar
19 We assume that we are given the Cartesian coordinates, x, y, and φ and we want to find anaytica expressions for the joint anges θ, θ, and θ in terms of the Cartesian coordinates. Substituting (4c) into (4a) and (4b) we can eiminate θ so that we have two equations in θ and θ : ( θ + θ ) ( θ + θ ) x (d) cosφ = cosθ + cos sin φ = sin θ + sin y (e) where the unknowns have been grouped on the right hand side; the eft hand side depends ony on the end effector or Cartesian coordinates and are therefore known. Rename the eft hand sides, x = x  cos φ, y = y  sin φ, for convenience. We regroup terms in (d) and (e), square both sides in each equation and add them: ( x cosθ ) = ( cos( θ + θ )) + ( y sin θ ) = ( sin( θ + θ )) After rearranging the terms we get a singe noninear equation in θ : ( x ) cosθ + ( y ) sin θ + ( x + y + ) 0 = (f) Notice that we started with three noninear equations in three unknowns in (ac). We reduced the probem to soving two noninear equations in two unknowns (de). And now we have simpified it further to soving a singe noninear equation in one unknown (f). Equation (f) is of the type P cosα + Q sinα +R = 0 (g) Equations of this type can be soved using a simpe substitution as shown in Appendix. There are two soutions for θ given by: where, θ = γ + σcos ( x + y + ) x + y (h) γ = a tan x y + y, x x + y, and σ = ±. Robot Geometry and Kinematics 9 V. Kumar
20 Note that there are two soutions for θ, one corresponding to σ=+, the other corresponding to σ=. Substituting any one of these soutions back into Equations (d) and (e) gives us: cos sin ( θ + θ ) ( θ + θ ) x cosθ = y sin θ = This aows us to sove for θ using the atan function in Appendix : θ y sin θ = atan x cosθ, θ Thus, for each soution for θ, there is one (unique) soution for θ. Finay, θ can be easiy determined from (c): (i) θ = φ  θ  θ (j) Equations (hj) are the inverse kinematics soution for the R manipuator. For a given end effector position and orientation, there are two different ways of reaching it, each corresponding to a different vaue of σ. These different configurations are shown in Figure 4. REFERENCE POINT (x,y) φ σ =+ σ = Figure 4 The two inverse kinematics soutions for the R manipuator: ebowup configuration (σ=+) and the ebowdown configuration (σ= ) Commanding a robot to move the end effector to a certain position and orientation is ambiguous because there are two configurations that the robot must choose from. From a Robot Geometry and Kinematics 0 V. Kumar
21 practica point of view, if the joint imits are such that one configuration cannot be reached this ambiguity is automaticay resoved. 5.5 Veocity anaysis When controing a robot to go from one position to another, it is not just enough to determine the joint and end effector coordinates of the target position. It may be necessary to continuousy contro the trajectory or the path taken by the robot as it moves toward the target position. This is essentia to avoid obstaces in the workspace. More importanty, there are tasks where the trajectory of the end effector is critica. For exampe, when weding, it is necessary to maintain the too at a desired orientation and a fixed distance away from the workpiece whie moving uniformy 4 aong a desired path. Thus one needs to contro the veocity of the end effector or the too during the motion. Since the contro action occurs at the joints, it is ony possibe to contro the joint veocities. Therefore, there is a need to be abe to take the desired end effector veocities and cacuate from them the joint veocities. A this requires a more detaied kinematic anaysis, one that addresses veocities or the rate of change of coordinates in contrast to the previous section where we ony ooked at positions or coordinates. Consider the R manipuator as an exampe. By differentiating Equation (4) with respect to time, it is possibe to obtain equations that reate the the different veocities. x = y = φ = θ s ( θ + θ ) s ( θ + θ + θ ) θ c + ( θ + θ ) c + ( θ + θ + θ ) c ( θ + θ + θ ) where we have used the short hand notation: s s = sin θ, s = sin (θ + θ ), s = sin (θ + θ + θ ) c = cos θ, c = cos (θ + θ ), c = cos (θ + θ + θ ) θ i denotes the joint speed for the ith joint or the time derivative of the ith joint anges, and x, y, and φ are the time derivatives of the end effector coordinates. Rearranging the terms, we can write this equation in matrix form: This is true of the human arm. If you consider panar movements, because the human ebow cannot be hyper extended, there is a unique soution for the inverse kinematics. Thus the centra nervous system does not have to worry about which configuration to adopt for a reaching task. 4 In some cases, a weaving motion is required and the trajectory of the too is more compicated. Robot Geometry and Kinematics  V. Kumar
22 Robot Geometry and Kinematics  V. Kumar ( ) ( ) ( ) ( ) θ θ θ = φ c c c c c c s s s s s s y x () The matrix is caed the Jacobian matrix 5 and we wi denote it by the symbo J. If you ook at the eements of the matrix they express the rate of change of the end effector coordinates with respect to the joint coordinates: θ φ θ φ θ φ θ θ θ θ θ θ = y y y x x x J Given the rate at which the joints are changing, or the vector of joint veocities, θ θ θ = q, using Equation (), we can obtain expressions for the end effector veocities, φ = y x p. If the Jacobian matrix is non singuar (its determinant is non zero and the matrix is invertibe), then we can get the foowing expression for the joint veocities in terms of the end effector veocities: p J q Jq p, = = () Thus if the task (for exampe, weding) is specified in terms of a desired end effector veocity, Equation () can be used to compute the desired joint veocity provided the Jacobian is non singuar: 5 The name Jacobian comes from the terminoogy used in mutidimensiona cacuus.
23 des θ s θ θ = ( s + s + s) ( s + s) ( c + c + c ) ( c + c ) c Naturay we want to determine the conditions under which the Jacobian becomes singuar. This can be done by computing the determinant of J and setting it to zero. Fortunatey, the expression for the determinant of the Jacobian, in this exampe, can be simpified using trigonometric identities to: J = sin θ () This means that the Jacobian is singuar ony when θ is either 0 or 80 degrees. Physicay, this corresponds to the ebow being competey extended or competey fexed. Thus, as ong we avoid going through this configuration, the robot wi be abe to foow any desired end effector veocity. x y φ des 5.6 Appendix 5.6. The ambiguity in inverse trigonometric functions Inverse trigonometric functions have mutipe vaues. Even within a 60 degree range they have two vaues. For exampe, if y = sin x the inverse sin function gives two vaues in a 60 degree interva: sin  y = x, πx Of course we can add or subtract π from either of these soutions and obtain another soution. This is true of the inverse cosine and inverse tangent functions as we. If y = cos x, the inverse cosine function yieds: cos  y = x, x Simiary, for the tangent function y = tan x, Robot Geometry and Kinematics  V. Kumar
24 the inverse tangent function yieds: tan  y = x, π +x This mutipicity is particuary troubesome in robot contro where an ambiguity may mean that there is more than one way of reaching a desired position (see discussion on inverse kinematics). This probem is circumvented by defining the atan function which requires two arguments and returns a unique answer in a 60 range. The atan function takes as arguments the sine and cosine of a number and returns the number. Thus if, s = sin x; c = cos x the atan function takes s and c as the argument and returns x: atan (s, c) = x The main idea is that the additiona information provided by the second argument eiminates the ambiguity in soving for x. To see this consider the simpe probem where we are given: s = ; c = and we are required to sove for x. If we use the inverse sine function and restrict the answer to be in the interva [0, π), we get the resut: x = sin  = π 6, 5π 6 Since we know the cosine to be, we can quicky verify by taking cosines of both candidate soutions that the first soution is correct and the second one is incorrect. cos π 6 = ; cos 5π 6 =  The atan function goes through a simiar agorithm to figure out a unique soution in the range [0,π). atan (, ) = π 6 The atan function is a standard function in most C, Pasca and Fortran compiers. Robot Geometry and Kinematics 4 V. Kumar
25 5.6. Soution of the noninear equation in (g) P cosα + Q sinα +R = 0 (g) Define γ so that P Q cos γ = P +Q and sin γ = P +Q Note that this is aways possibe. γ can be determined by using the atan function: γ = a tan P Q + Q, P P + Q Now (g) can be rewritten as: or cos γ cos α + sin γ sin α + cos ( α γ) = P R + Q P R + Q = 0 This gives us two soutions for α in terms of the known ange γ: α = γ + σcos P R + Q, σ = ± Robot Geometry and Kinematics 5 V. Kumar
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