We describe the kinematics of X-Y pedestals with unideal geometry caused by joint and pointing mis- alignment that might arise from issues in manufac- turing, assembly or mechanical wear. We present it's real and complex inverse kinematic solutions that are produced through classical methods of analysis and test their validity using simulations of discrete rotations. Introduction X-Y pedestals are manipulators of two revolute joints ideally positioned orthogonally with one stacked on top of the other, with both parallel to the tangent plane of Earth spheroid at that location. They are a popular design choice for low- earth-orbit satellite tracking applications of high- elevation passes. They are preferred over the al- ternative altitude-azimuth pedestals that su er the zenith problem at positions closer to the azimuth singularity [1]. While all serial manipulators have singularities [2], For X-Y pedestals the singularity is located at the axis of the stationary joint. This is a concern only for low elevation applications which is a rare occurence for satellite communica- tions[3]. X-Y pedestal kinematics is not as intuitive as their altitude-azimuth counterpart, but given their popularity previous research delving into their dynamics, kinematics and control that lessens this property already exists[4]. In this paper we hope to ll the gap for a more general inverse kinematics solution that considers pedestals with misalignment ∗E-mail: bugra.coskun@profen.com †ORCID: 0009-0002-0177-684Xissues without the need for resynthesizing the in- verse kinematics solution. xyz wˆlˆp αλβγ Figure 1: Pedestal coordinate system with exagerated misalign- ments For the mathematical model; the stationary joint axis will be deemed the X-joint and will coincide with ^xin the pedestal standard basis and it's rotation angle shall be expressed as . For = 0 the Y-joint axis will be ^ wat an angle with ^x and let be it's rotation angle. ^ wwill belong to the same plane de ned by the vectors ^ xand ^y. As such the joint misalignment is 2. Let ^pbe the orientation vector at = 0 and = 0. Similar to joint misalignment, pointing misalignment is  2 . In a real-world case the pedestal basis should be de ned in the NWU or NED frame so as to meet these de nitions. This involves calculating the pedestal ^ zfrom positions of pedestal ^ xand ^wvectors in the selected environment frame and produce a rotation matrix. This will make it so that any joint misalignment can be considered using the 1 de nitions above. Let the target orientation be ^l. For inverse kinematics this is usually given as azimuth and elevation angles and in left-handed spherical coordinates. Taking the negative of the azimuth angle gives us and we can produce the cartesian vector using the appropriate spherical to cartesian mapping. Forward Kinematics We rst consider the rotation around the Y-joint, as Y-joint axis is modi ed by the X-joint rotation, this will save us from calculating the new axis of the Y-joint. This is an example of extrinsic rotation. ^w=2 4cos sin 03 5; (1) The skew-symmetric cross product matrix of ^ wis therefore: W=2 40 0 sin  0 0cos sincos 03 5; (2) The rotation matrix for Y-joint in SO(3) group is given by the Rodrigues' rotation formula: R^w( ) =I+ (sin )W+ (1cos )W2;(3) The elementary rotation matrix around the ^ xaxis is in the form: Rx( ) =2 41 0 0 0 cos sin 0 sin cos 3 5; (4) We de ne the orientation vector at initial con gu- ration = 0 and = 0 to be: ^p=2 4coscos sincos sin 3 5; (5) Thus the forward kinematic equation becomes: ^l=R^x( )R^w( )^p: (6) The resulting line-of-sight vector is in the pedestal frame, which ideally shall coincide with the North- West-Up frame for our model. Any tilt in the pedestal frame can be accounted for if we know the orientation of the pedestal and produce the rotation matrix that gives us the position of pedestal basis in the NWU frame similar to gimbal calculations for stabilizing the line of sight[5].Inverse Kinematics For inverse kinematics we introduce the concepts of joint surface and loose reachability. We de ne the surface of a joint as being the set of all unitary vectors where their dot products against the joint axis are some value c. c2(1;1);S^k(c) =f^v: ^v^k=cg: (7) For any rotation around the joint axis the dot products of vectors belonging to the surface are conserved: 8^v2S^k(c);82(;]; R^k()^v2S^k(c)*R^k()2SO(3):(8) Following that, ^l2S^x(^l^x))R^x( )R^w( )^p=^l (9) ()R^w( )^p2S^x(^l^x) (10) ()R^w( )^p^x=^l^x: (11) Thus if9 2(;] such that Eq(9) holds we consider the position to be loosely reachable (ie. a real solution exists without considering joint limitations). Solving Eq(10) for we get: = sin1cos cos cos cos sin sin : (12) If the position is not loosely reachable and sin1(x) is extended into the complex plane so that it's de ned forjjxjj>1 then Im( )6= 0. In such a case the complex angles can be used to nd the closest approximate position in the workspace [6]. For solving X-Joint angle independently from the angle we leverage the fact that ^ pshall belong to both joint surfaces if a proper angle exists, ^p=R^w( )^p; (13) ^p2S^x(^l^x)\S^w(^p^w); (14) Using this this property we construct the equation that shall consider in isolation from , R^x( )^p=^l() ^p=R1 ^x( )^l (15) ()R^x( )^l2S^w(^p^w) (16) ()R^x( )^l^w= ^p^w; (17) 2 From Eq(15) we get the transcendental equation: cos sin cos sin+ sin sin sin cos cos cos+ cos = 0;(18) To solve this we employ Euler's formula and con- sider in the complex plane. With some algebraic manipulation and application of the quadratic for- mula we receive the equation for the nearest root: =ilog0 BBBBBB@(cos2 cos2 2 cos cos cos cos + cos2 + cos21)1 2 + cos cos coscos (sin cos isin ) sin1 CCCCCCA:(19) Alternatively (as in some computational cases it may be undesirable to use complex numbers), through vector analysis we can compare ^land ^p in the plane de ned by ^ xusing their cross products with ^xand synthesize another equation that way. Incidentally we do not need to project ^land ^p entirely, their rotation angle around the joint axis will be preserved for their cross product with the joint axis. 8c2(1;1);8^k;8^v;8^u2S^k(c); (20) R^k()^v= ^u()R^k()(^v^k) = (^u^k) (21) It is clear that the resulting vector from the cross product of normalized ^ p^xand ^p^xvectors will be parallel to ^ x. And will be equivalent if the rotation angle is positive and vice versa. To nd the sign of the angle in left-handed rotation around ^xwe use the dot product. Thus the vector equation and it's simpli ed trigonometric form are constructed as such: = sin1" ^p^x jj^p^xjj^l^x jj^l^xjj! ^x# ; = sin10 BBBBBBBBBBBB@(sin sin cos sincos ) sin + sin sin cos cos (cos2 cos2 + 1)1 2 [(sin sin cos sincos )2 + sin2 cos2 ]1 21 CCCCCCCCCCCCA:(22)Simulation of Discrete Rotations To verify the produced equations we test their output by comparing the result of rotations acting on ^pto the target orientation ^l. To visualize the process further we rotate ^ paround ^w rst to produce ^pand then apply the rotation around ^ x. Figure 2: Simulation of discrete rotations with produced angles For the simulation in Figure 2, we apply the parameters = 4:5; = 4; = 2:25; = 2:25 in radians and we use the complex equation for calculating the angle. Inverse kinematic results were 0:556 and 0:796 +i6:287e-17. We take the real part of . Rotated ^ pand ^lwere identical and no error was calculated. Simulations show that for 0 <"<< 1 (for the simulation code using Sympy it's been observed that "1e-9): jj^l^xjj<1" (23) !^l(^R^x( )R^w( )^p)>1" (24) Thus for non-singular values the simulation did not show any signs of signi cant error, as well as for negative elevation values and the zenith. The simulation has been written in Python using Sympy and Matplotlib. And angles produced has been cross checked using Octave. 3 Conclusion Using analytical methods, we produced and, through simulations, veri ed a more general inverse kinematics solution for X-Y pedestals that take into account joint and pointing misalignments. The inverse kinematics equations have a relatively low computational complexity and (if the computing environment supports complex numbers) are robust for out-of-workspace positions. Acknowledgements This research has been funded by Profen Com- munication Technologies & Services, Inc. The author is thankful to Mustafa C  elik, Cem Do gan and Kadir Koca for their guidance on X- Y pedestals, to Merve G uven c for the inspiring conversations and to Rzk Tekay for being the catalyst for this research.