###### Abstract  
As for industrial robots' point-to-point joint motion planning with constrained velocity, cubic polynomial planning has the problem of discontinuous acceleration; quintic polynomial planning requires acceleration to be specified in advance, which will likely cause velocity to fluctuate largely because appropriate acceleration assigned in advance is hardly acquired. Aiming at these problems, a modified cubic Hermite interpolation for joint motion planning was proposed. In the proposed methodology, knots of cubic Hermite interpolation need to be reconfigured according to the initial knots. The formulas for how to build new knots were put forward after derivation. Using the newly-built knots instead of initial knots for cubic Hermite interpolation, joint motion planning was carried out. The purpose was that the joint planning not only satisfied the displacement and velocity constraints at the initial knots but also guaranteed C\({}^{2}\) continuity and less velocity fluctuation. A study case was given to verify the rationality and effectiveness of the methodology. Compared with the other two planning methods, it proved that the raised problems can be solved effectively via the proposed methodology, which is beneficial to the working performance and service life of industrial robots. industrial robot; point-to-point motion planning; velocity constraint; second-order continuity; Hermite interpolation; knot reconfiguration +
Footnote †: journal: Applied Sciences  
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Joint Motion Planning of Industrial Robot Based on Modified Cubic Hermite Interpolation with Velocity Constraint  
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Y. ShiTherefore, motion planning in joint space can reduce inverse kinematics calculation largely so that computation efficiency is improved, and also can avoid joints singularity and redundancy [6]. As a result, motion planning in joint space is more suitable for the point-to-point (PTP) motion of industrial robots. The main methods of joint motion planning are reviewed below. Linear planning with parabolic transition has the advantage that joint speed remains constant in the linear phase, but sufficient transition time must be ensured, and angular acceleration is not continuous. The widely used cubic polynomial planning has the advantages of simple principle and less computation. However, when it is used to multiple points with constrained velocities, the angular acceleration at the connection points is broken, which will lead to vibrations and impacts and make the robot arm tremble. Quintic polynomial planning has continuous angular acceleration. However, quintic polynomial interpolation requires specifying in advance the angular acceleration value of the joint at each target point. If it is not preset properly, the angular velocities will fluctuate back and forth, which is not conducive to joint motion control. Frustratingly, it is difficult to give appropriate angular acceleration values in advance [7]. In addition, quintic multinomial has a higher degree than cubic polynomial so that velocities and accelerations change more intensely, and larger velocity and acceleration peaks appear. B spline interpolation is also used in motion planning [8; 9; 10]. However, when the cubic B-spline curve is used for multiple points with constrained velocities, acceleration is not continuous at connecting points as well. Some researchers carried out lots of studies about cubic B spline and realized G\({}^{2}\) continuity at the connecting knots [11; 12]. However, G\({}^{2}\) continuity refers to curvature continuity, and continuity of second-order derivative C\({}^{2}\) is not fully achieved. In order to solve the complicated joint planning with multiple points further, [PERSON] proposed a 3-5-3 polynomial splicing method [13]. This method allows to specify angular velocity at the intermediate target points arbitrarily, and angular acceleration is continuous. However, when there are more than 5 knots, angular acceleration at the target knot needs to be given in advance for quintic polynomial.