A systematic robot labelling notation widely used is the Denavit-Hartenberg parameters [11]. Two angles and two distances allow establishing recurrence. Fig. 1.a shows the three links that comprises the representation of the 2-DOF OSPS: link 0 or grounded element, link 1 for azimuth motion, and link 2 for elevation motion.

Fig. 2 depicts the azimuth and elevation motion of the panel. Each element has a reference system, Fig. 1.b: fixed (inertial) {x₀-y₀-z₀} or {0}, movable reference system {x₁-y₁-z₁} or {1} for the azimuth motion, and local reference system {x₂-y₂-z₂} or {2} for the elevation motion. The latter element is attached to the panel and the securing structure elements.

Fig. 1.b also depicts the Denavit-Hartenberg parameters. Joint angle parameter θ_i is the angle that element i-th must be rotated about the z_i-1 axis in order to align axes X_i-1 and xi in the same plane, while axis y_i-1 and yi have the same direction. Distance d_i is the offset distance along the z_i-1 axis from the origin of the (i-1)-th system {i-1} to the origin of the system {i}. Distance ai is the offset distance along the x_i-1 axis from the origin of system {i-1} to the origin of the system {i}. Angle α_i is the angle about the x_i-1 axis that the must be rotated the system {i-1} to get aligned the axes z_i-1 and z_i, after turning the angle θ_i . The recurrence in the parameters leads to a transformation matrix, ^i-1A_i, as the result of successive standard transformations to relate two consecutive coordinate axes {i-1} and {i}, given by (1) [12], where c_i and s_i are the cosine and sine of θ_i, respectively; and c_αi and s_αi are the cosine and sine of the link angle α_i, respectively. Table 1 presents the numerical values [13,14] of the Denavit-Hartenberg parameters of the OSPS in Fig. 1.b. From the recurrence in (1) is possible to relate the coordinate systems {0}, {1} and {2} to yield the transformation matrix of the OSPS that relates the second element with the ground, as presented in (2) [11,12,14]. For the case in this report, the parameter values are d₁=355mm, d₂=0mm, a₁=0mm, a₂=91mm, α₁=90°, α₂=0°.

Stating the dynamic equation of a manipulator allows relating applied forces at the actuated joints with the expected motion [15]. In the direct dynamics analysis, the equation of motion is used to find the response of the robot arm to some applied forces. In the inverse dynamics problem, this equation is used to find the actuator torques and/or forces required to generate a desired path at the tool [12,15]. Three approaches are used for establishing the dynamic equation of a manipulator: Newton-Euler, virtual work principle, and Lagrange formulation [12,15,16]. The dynamic model of serial manipulators is found in a simpler way by the Lagrangian method [16]. The general formulation presented in [16] is adapted next for the 2-DOF serial manipulator that represents the OSPS. The Langrange function, L, of the robot is found by (3), with K: kinetic energy, U: potential energy of the manipulator (OSPS). Equation (4) shows the time evolution of the OSPS as a function of the n generalized coordinates, with qi: i-th generalized coordinate, Q_i: i-th generalized force of the system (i=1,2 since there are 2 revolute joints) or torques applied at each joint by the motors. Equation (5) describes the dynamics of the system, with τ: applied torque vector at the joints, M: inertia matrix, : angular acceleration vector, V: centrifugal and Coriollis forces vector, and G: torque due to the gravity effect. The kinetic energy of the i-th element, K_i, is given by (6), where the contribution of both center of mass linear velocity, v_ci, and angular velocity, ω_i, of i-th element is considered with m_i: mass of the element, and I_i: inertia matrix of the element with respect to its own center of mass. Term Ii is found by (7), where ⁰R_i is the orientation matrix of the element as seen if the fixed coordinate system {0} given by the 3x3 submatrix [(1,1)…(3,3)] of the transformation matrix in (2); mi is the mass of the element; and ⁱ_Ii is the inertia tensor in its own center of mass, as seen in its own coordinate system {i}, as seen in (8).

The total velocity vector of the center of mass of the i-th element, , given by (9), is related with the generalized coordinates velocity vector, , by the Jacobian matrix of the element, J_i, as seen in (10). The 6x1 size Jacobian matrix of the element is created from the two 3x1 submatrices, see (11), J_vi and J_ωi that represent the center of mass linear velocity Jacobian and angular Jacobian matrixes of the element, respectively. Submatrices J_vi and J_ωi for revolute joints are found from the recurrence in (12) and (13), respectively, where z_j-1: unit vector for the direction of the rotation axis of the element j-1, as seen in {0}; ^j-1p_ci: position vector from the origin of {j-1} to the center of mass of the i-th element, and the “x” in (12) indicates the cross product operation. The size of submatrices J_vi and J_ωi is 3xn, when j>i these entities are completed with [0]_1xn as seen in (14) and (15), respectively, where n is the number of generalized coordinates. The inertia matrix, M, of the system is found with (16). The total potential energy of the OSPS, U, is also found with (17), where g^T; column gravity vector with g=9.81m/s².

The i-th term of the centrifugal and Coriolis forces vector, V_i, is obtained with (18), where the forces vector is given in (19). The torque contribution of each element’s weight, G_i, yields (20), the total vector is also given in (21). Similarly, the external torque elements, τ_i, let yielding the external forces and torques vector τ, in (22). By substituting (6) to (17) into (3) and solving into (4), it is obtained the dynamic equation of the manipulator. For the inverse dynamics, it is solved from (5) to yield (23). By following the presented formulation it is obtained the dynamic model for the OSPS in (24). The attained model is non-linear and the torque applied by the motors in both joints will vary along the motion. It is also apparent from the same equation that the inertia to be overcome by the motor at joint 2 is constant. However, inertia load for the motor at joint 1 will change. Since both joints are revolute type (no prismatic joint), there are not Coriolis terms in the model but centrifugal force terms. Due to the design of the OSPS, only the weight of the second element is contributing to the gravity term.