!"#$Introduction Robotics, lecture 3 of 9drDraganKosti!WTB Dynamics and ControlMay-June 2008Introduction Robotics%"#$Introduction Robotics, lecture 3 of 9 Recapitulation Forward kinematics of RRR robot Inverse kinematics problemOutline#"#$Introduction Robotics, lecture 3 of 9Recapitulation&"#$Introduction Robotics, lecture 3 of 9Robot manipulatorsKinematic chain is series of links and joints.Types of joints:rotary (revolute, )prismatic (translational, d).SCARAgeometryschematic representations of robot joints'"#$Introduction Robotics, lecture 3 of 9Common geometries of robot manipulators$"#$Introduction Robotics, lecture 3 of 9Forward kinematics problemDetermine position and orientation of the end-effectoras function of displacements in robot joints.("#$Introduction Robotics, lecture 3 of 9DH convention for homogenous transformations (1/2)An arbitrary homogeneous transformation is based on 6independent variables: 3 for rotation + 3 for translation.DH convention reduces 6 to 4, by specific choice ofthe coordinate frames.In DH convention, each homogeneous transformation has the form:)"#$Introduction Robotics, lecture 3 of 9DH convention for homogenous transformations (2/2)Position and orientation ofcoordinate frame iwith respect to frame i-1is specified by homogenous transformation matrix:aiqiq0qiqi+1x0xi-1xizizi-1xny0ynz0zndii0nwhere*"#$Introduction Robotics, lecture 3 of 9Physical meaning of DH parametersLink length aiis distance from zi-1to zimeasured along xi.Link twist iis angle between zi-1and zimeasured in plane normal to xi(right-hand rule).Link offset diis distancefromoriginof frame i-1to the intersectionxiwithzi-1, measuredalongzi-1.Joint angleiis angle from xi-1 to ximeasuredin planenormalto zi-1(right-hand rule).aiqiq0qiqi+1x0xi-1xizizi-1xny0ynz0zndii0n!+"#$Introduction Robotics, lecture 3 of 9DH convention to assign coordinate frames1.Assign zito be the axis of actuation for joint i+1(unless otherwise stated zncoincides with zn-1).2.Choose x0and y0so that the base frame is right-handed.3.Iterative procedure for choosing oixiyizidepending on oi-1xi-1yi-1zi-1(i=1, 2, , n-1):a) zi1and ziare not coplanar; there is an unique shortest line segment fromzi1to zi, perpendicular to both; this line segment defines xiand the point where the line intersects ziis the origin oi; choose yito form a right-handed frame,b) zi1is parallel to zi; there are infinitely many common normals; choose xiasthe normal passes through oi1; choose oias the point at which this normalintersects zi; choose yito form a right-handed frame,c) zi1intersects zi; axis xiis chosen normal to the plane formed by ziand zi1;its positive direction is arbitrary; the most natural choice of oiis theintersection of ziand zi1, however, any point along the zisuffices;choose yito form a right-handed frame.!!"#$Introduction Robotics, lecture 3 of 9Case-study: RRR robot manipulatorx0q1q2-q3x1x2x3y0y1y2y3z01d1d2a2a3d3z1z2z3waistshoulderelbow1 - twist angleai - link lenghtsdi - link offsetsqi - displacements!%"#$Introduction Robotics, lecture 3 of 9DH parameters of RRR robot manipulator!#"#$Introduction Robotics, lecture 3 of 9Forward kinematics of RRR robot manipulator (1/2)Coordinate frame o3x3y3z3is related with the base frame o0x0y0z0via homogenous transformation matrix:!" #$% &===1310303321030(q)x(q)(q)(q)A(q)AA(q)TRwhereTqqq][321=qTzyx][)(03=qx]000[31=0!&"#$Introduction Robotics, lecture 3 of 9Forward kinematics of RRR robot manipulator (2/2),,Position of end-effector:[]132223231)sin(cos)cos(cosqddqaqqaqx++++=[]132223231)cos(cos)cos(sinqddqaqqaqy+++=122323sin)sin(dqaqqaz+++=Orientation of end-effector:! ! !" #$ $ $% &++++++=0)cos()sin(cos)sin(sin)cos(sinsin)sin(cos)cos(cos32321321321132132103qqqqqqqqqqqqqqqqqqR!'"#$Introduction Robotics, lecture 3 of 9Inverse Kinematics!$"#$Introduction Robotics, lecture 3 of 9Inverse kinematics problem Inverse kinematics (IK): determine displacements in robot joints that correspond to given position and orientation of the end-effector.!("#$Introduction Robotics, lecture 3 of 9Illustration: IK for planar RR manipulator (1/2) Elbow down IK solution!)"#$Introduction Robotics, lecture 3 of 9Illustration: IK for planar RR manipulator (2/2)Elbow up IK solution!*"#$Introduction Robotics, lecture 3 of 9The general IK problem (1/2)Given a homogenous transformation matrix HSE(3)find (multiple) solution(s) q1,,qnto equationHere, H represents the desired position and orientation of the tip coordinate frame onxnynznrelative to coordinate frame o0x0y0z0of the base; T0nis product of homogenous transformation matrices relating successive coordinate frames:%+"#$Introduction Robotics, lecture 3 of 9The general IK problem (2/2)Since the bottom rows of both T0nand Hare equal to [0 0 0 1], equationgives rise to 4 trivial equations and 12 equations in nunknowns q1,,qn:Here, Tijand Hijare nontrivial elements of T0nand H.%!"#$Introduction Robotics, lecture 3 of 9Illustration: Stanford manipulator%%"#$Introduction Robotics, lecture 3 of 9FK of Stanford manipulator%#"#$Introduction Robotics, lecture 3 of 9Example of IK solution for Stanford manipulator Rotational equations (correspond to R06): Positional equations (correspond to o06): One solution:%&"#$Introduction Robotics, lecture 3 of 9Nature of IK solutionsFK problem has always unique solution whereas IK problem may or may not have a solution; if IK solution exists, it may or maynot be unique; solving IK equations, in general, is much too difficult.It is preferable to find IK solutions in closed-form:faster computation (e.g. at sampling time of 1 [ms]),if multiple IK solutions exist, then closed-form allows us to develop rules for choosing a particular solution among several.Existence of IK solutions depends on mathematical as well as engineering considerations.We assume that the given position and orientation is such that at least one IK solution exists.%'"#$Introduction Robotics, lecture 3 of 9Kinematic decoupling (1/3)General IK problem is difficult BUTfor manipulators having 6 joints with the last 3 joint axes intersecting at one point, it is possible to decouple the general IK problem into two simpler problems:inverse position kinematics and inverse orientation kinematics.IK problem: for given Rand osolve 9rotational and 3 positional equations:%$"#$Introduction Robotics, lecture 3 of 9Kinematic decoupling (2/3)Let ocbe the intersection of the last 3 joint axes; as z3, z4, and z5intersect at oc, the origins o4and o5will always be at oc;the motion of joints 4, 5 and 6 will not change the position of oc;only motions of joints 1, 2 and 3 can influence position of oc.Spherical wrist as paradigm.%("#$Introduction Robotics, lecture 3 of 9Kinematic decoupling (3/3)'q1, q2, q3''q4, q5, q6%)"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: inverse position problemInverse tangent function Atan2(xc,yc)is defined for all (xc,yc)(0,0)and equals the unique angle 1such that:,cos221cccyxx+=.sin221cccyxy+=%*"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: left arm configuration**r#+"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: right arm configuration+**r#!"#$Introduction Robotics, lecture 3 of 9Law of cosines:*rArticulated manipulator: IK solution for 3+elbow down; -elbow up#%"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: IK solution for 2 1 22=1-21=Atan2(r,s)2=Atan2(a2+a3cos3,a3sin3)##"#$Introduction Robotics, lecture 3 of 9Four IK solutions 1- 3for articulated manipulatorPUMA robot as an example ofthe articulated geometry.#&"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: inverse orientation problem Equation to solve:#'"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: IK solutions for 4and 5Equations given by the third column in :If not both right-hand sides of the first two equations are zero:If positive square root is chosen in solution for 5:#$"#$Introduction Robotics, lecture 3 of 9Articulated manipulator: IK solutions for 6The first two equations given by the last row in :Analogous approach if negative square root is chosen in solution for 5.-s5c6= s1r11-c1r21s5s6= s1r12-c1r22From these equations it follows: