2. Elliptic jerk motion profile

A dimensionless formulation is adopted for the sake of generality. For given displacement length h and motion duration T:

• the dimensionless time is the time t normalized by T: t _ad = t/T;

• the dimensionless position is the position s normalized by h: s _ad = s/h;

• the dimensionless velocity is the velocity v normalized by h/T: v _ad = vT/h;

• the dimensionless acceleration is the acceleration a normalized by h/T ²: a _ad = aT ²/h;

• the dimensionless jerk is the jerk j normalized by h/T ³: j _ad = jT ³/h.

The proposed motion law is based on the jerk profile shown in Fig. 1 as a function of the dimensionless time, which varies from 0 to 1 from rest to rest. This profile is divided into seven phases (#1–#7, Fig. 1). The ith phase starts at t _ad,i-1 and ends at t _ad,i.

Figure 1. Elliptic jerk motion profile as a function of the dimensionless time t _ad.

The first three phases are characterized by positive acceleration and have an overall duration t _ad,pa. The fourth phase has null acceleration and constant velocity. The last three phases are characterized by negative acceleration and have an overall duration t _ad,na. Consequently, the duration of the fourth phase is (1 − t _ad,pa − t _ad,na).

The positive acceleration motion is performed with positive jerk in phase #1, with null jerk in phase #2, and with negative jerk in phase #3. The durations of the phases #1 and #3 are, respectively, t _ad,papj and t _ad,panj. Consequently, the duration of phase #2 is (t _ad,pa − t _ad,papj − t _ad,panj).

The deceleration is performed with negative jerk in phase #5, with null jerk in phase #6, and with positive jerk in phase #7. The durations of the phases #5 and #7 are, respectively, t _ad,nanj and t _ad,napj. Consequently, the duration of phase #6 is (t _ad,na − t _ad,nanj − t _ad,napj).

There are four phases with non-null jerk: #1, 3, #5, and #7. In these phases, the jerk profile is elliptic. The two phases with positive jerk, #1 and #7, have, respectively, maximum values of +j _ad1 and +j _ad7. The two phases with negative jerk, #3 and #5, have, respectively, minimum values of −j _ad3 and −j _ad5. With the hypothesis of the elliptical profile, the dimensionless jerk law is completely defined by 10 positive parameters:

• four positive jerk peak parameters: j _ad1, j _ad3, j _ad5, j _ad7;

• six time parameters: t _ad,pa, t _ad,na, t _ad,papj, t _ad,panj, t _ad,nanj, t _ad,napj.

3. Analytical expression of the motion profile

3.1. Expression of the jerk profile

The analytical expression of the motion profile can be derived from the representation of the semi-elliptical shape of the jerk of a generic phase shown in Fig. 2.

Figure 2. Elliptic jerk as a function of dimensionless time t _ad for the ith phase.

The parametric equation of the semi-ellipse of Fig. 2 is

(1) \begin{equation} \begin{array}{l} j_{\mathrm{ad}}=\sigma j_{\mathrm{ad},\mathrm{i}}sen\!\left(u\right)\\[9pt] \cos\!\left(u\right)=1-\dfrac{2\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right)}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\\[12pt] \sigma _{i}=\left\{\begin{array}{l} +1,i=1,7\\[5pt] -1,i=3,5\\[5pt] 0,i=2,4,6 \end{array}\right. \end{array} \end{equation}

where 0 ≤ u ≤ π. By means of some trigonometric manipulation, we can obtain the following expression of the jerk, valid for all the phases, including the null jerk ones, for which σ _i = 0:

(2) \begin{equation} j_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)=2\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\sqrt{\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}} \end{equation}

3.2. Expression of the acceleration profile

The expression of the acceleration profile has been found by analytical integration of Eq. (2), obtaining the following formula for the ith phase, for t _ad,i-1 ≤ t _ad < t _ad,i:

(3) \begin{align} a_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)&=a_{\mathrm{ad},\mathrm{i}-1}+\frac{1}{4}\!\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}a\cos \!\left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)+\nonumber \\[5pt] & \quad -\frac{1}{2}\!\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\!\left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)\sqrt{\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}} \end{align}

where a _ad,i-1 is the acceleration at the end of the i − 1th phase. In particular, for the phases with null jerk (i = 2, 4, 6), σ _i = 0, then acceleration is constant, and Eq. (3) becomes:

(4) \begin{equation} a_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)=a_{\mathrm{ad},\mathrm{i}-1} \end{equation}

3.3. Expression of the velocity profile

The expression of the velocity profile has been found by analytical integration of Eq. (3), obtaining the following formula for the ith phase, for t _ad,i-1 ≤ t _ad < t _ad,i:

(5) \begin{align} v_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)&=v_{\mathrm{ad},\mathrm{i}-1}+a_{\mathrm{ad},\mathrm{i}-1}\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right)+\nonumber\\[5pt] & \quad -\frac{1}{8}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{2}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\!\left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)\arccos \left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)+\nonumber\\[5pt] & \quad +\frac{1}{4}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{2}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\sqrt{\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}}+\nonumber \\[5pt] & \quad -\frac{1}{3}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{2}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\sqrt{\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}\right)^{3}} \end{align}

where v _ad,i-1 is the velocity at the end of the i − 1th phase. In particular, for the phases with null jerk (i = 2, 4, 6), σ _i = 0, then acceleration is constant, and the velocity profile becomes:

(6) \begin{equation} v_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)=v_{\mathrm{ad},\mathrm{i}-1}+a_{\mathrm{ad},\mathrm{i}-1}\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right) \end{equation}

3.4. Expression of the position profile

The expression of the velocity profile has been found by analytical integration of Eq. (5), obtaining the following formula for the ith phase, for t _ad,i-1 ≤ t _ad < t _ad,i:

(7) \begin{align} s_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)&=s_{\mathrm{ad},\mathrm{i}-1}+v_{\mathrm{ad},\mathrm{i}-1}\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right)+\frac{1}{2}a_{\mathrm{ad},\mathrm{i}-1}\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{2}+\nonumber \\[5pt] &\quad +\frac{5}{128}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{3}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\arccos \left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)+\nonumber \\[5pt] &\quad -\frac{5}{64}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{3}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\!\left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)\sqrt{\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}}+\nonumber\\[5pt] &\quad -\frac{1}{8}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{3}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\!\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}\right)\arccos \left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)+\nonumber\\[5pt] &\quad +\frac{1}{24}\left(t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{3}\sigma _{\mathrm{i}}\;j_{\mathrm{ad},\mathrm{i}}\!\left(1-2\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)\sqrt{\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}-\left(\frac{t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}}{t_{\mathrm{ad},\mathrm{i}}-t_{\mathrm{ad},\mathrm{i}-1}}\right)^{2}\right)^{3}} \end{align}

where s _ad,i-1 is the position at the end of the i − 1th phase. In particular, for the phases with null jerk (i = 2, 4, 6), σ _i = 0, and the position profile becomes:

(8) \begin{equation} s_{\mathrm{ad}}\!\left(t_{\mathrm{ad}}\right)=s_{\mathrm{ad},\mathrm{i}-1}+v_{\mathrm{ad},\mathrm{i}-1}\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right)+\frac{1}{2}a_{\mathrm{ad},\mathrm{i}-1}\!\left(t_{\mathrm{ad}}-t_{\mathrm{ad},\mathrm{i}-1}\right)^{2} \end{equation}

3.5. Dependence of the motion profile parameters

As discussed in Section 2, the jerk profile is completely defined by 10 parameters, four jerk peak parameters and six time parameters. Consequently, also the acceleration, velocity, and position profiles, which are obtained by successive integrations, are defined by these 10 parameters. Nevertheless, these parameters are not independent, once these conditions are imposed:

1. I. null acceleration at the end of phase #3: a _ad(t _ad3) = a _ad(t _ad,pa) = 0;

2. II. null acceleration at the end of phase #7: a _ad(t _ad7) = a _ad(1) = 0;

3. III. null velocity at the end of phase #7: v _ad(t _ad7) = v _ad(1) = 0;

4. IV. unit dimensionless position at the end of phase #7: s _ad(t _ad7) = s _ad(1) = 1.

Since acceleration is the integral of the jerk, conditions I and II mean that the semi-elliptical areas defined by the jerk profiles are equal for the couple of phases #1–#3, and for the couple #5–#7. The four conditions I–IV establish four equations in the 10 profile parameters, which can be used in different ways. One possible choice, which is adopted in the following of the work, is to obtain the four jerk peak parameters (j _ad1, j _ad3, j _ad5, j _ad7) as functions of the six time parameters (t _ad,pa, t _ad,na, t _ad,papj, t _ad,panj, t _ad,nanj, t _ad,napj). Using this approach, the equations representing the conditions I–IV have been elaborated, obtaining the following expressions of the four peak parameters as a function of the time parameters:

(9) \begin{equation} j_{\mathrm{ad}1}=\frac{1}{c_{1}+c_{2}+c_{3}} \end{equation}

(10) \begin{equation} j_{\mathrm{ad}3}=\frac{t_{\mathrm{ad},\text{papj}}}{t_{\mathrm{ad},\text{panj}}}j_{\mathrm{ad}1} \end{equation}

(11) \begin{equation} j_{\mathrm{ad}5}=c_{0}\frac{t_{\mathrm{ad},\text{papj}}}{t_{\mathrm{ad},\text{napj}}}j_{\mathrm{ad}1} \end{equation}

(12) \begin{equation} j_{\mathrm{ad}7}=\frac{t_{\mathrm{ad},\text{nanj}}}{t_{\mathrm{ad},\text{napj}}}j_{\mathrm{ad}5} \end{equation}

where the four constants c ₀–c ₃ are defined by the following equations:

(13) \begin{equation} c_{0}=\frac{2t_{\mathrm{ad},\mathrm{pa}}-t_{\mathrm{ad},\text{papj}}-t_{\mathrm{ad},\text{panj}}}{2t_{\mathrm{ad},\mathrm{na}}-t_{\mathrm{ad},\text{nanj}}-t_{\mathrm{ad},\text{napj}}} \end{equation}

(14) \begin{equation} c_{1}=\frac{\pi }{8}t_{\mathrm{ad},\text{papj}}t_{\mathrm{ad},\mathrm{pa}}\!\left(t_{\mathrm{ad},\mathrm{pa}}-t_{\mathrm{ad},\text{papj}}\right)+\frac{5}{128}\pi t_{\mathrm{ad},\text{papj}}\!\left(t_{\mathrm{ad},\text{papj}}^{2}-t_{\mathrm{ad},\text{panj}}^{2}\right) \end{equation}

(15) \begin{equation} c_{2}=\frac{\pi }{8}t_{\mathrm{ad},\text{papj}}\!\left(1-t_{\mathrm{ad},\mathrm{pa}}-t_{\mathrm{ad},\mathrm{na}}\right)\!\left(2t_{\mathrm{ad},\mathrm{pa}}-t_{\mathrm{ad},\text{papj}}-t_{\mathrm{ad},\text{panj}}\right) \end{equation}

(16) \begin{equation} c_{3}=\frac{\pi }{8}c_{0}t_{\mathrm{ad},\text{papj}}t_{\mathrm{ad},\mathrm{na}}\!\left(t_{\mathrm{ad},\mathrm{na}}-t_{\mathrm{ad},\text{napj}}\right)+\frac{5}{128}\pi c_{0}t_{\mathrm{ad},\text{papj}}\!\left(t_{\mathrm{ad},\text{napj}}^{2}-t_{\mathrm{ad},\text{nanj}}^{2}\right) \end{equation}

In particular, Eq. (10) represents the ratio between the two jerk peaks in acceleration due to condition I, while Eq. (12) represents the ratio between the two jerk peaks in deceleration due to condition II. Equations (9) and (11) derive, respectively, from conditions IV and III.

For example, Fig. 3 shows the jerk, acceleration, velocity, and acceleration motion profiles obtained imposing the time parameters t _ad,pa = 0.3, t _ad,na = 0.5, t _ad,papj = 0.05, t _ad,panj = 0.15, t _ad,nanj = 0.2, and t _ad,napj = 0.25. The resulting jerk peak parameters are j _ad1 = 208.07, j _ad3 = 69.36, j _ad5 = 37.83, and j _ad7 = 30.26. The dimensionless acceleration varies between a maximum of 8.17 in phase #2 and a minimum of −5.94 in phase #6, while the dimensionless velocity reaches a maximum value of 1.634 in phase #4.

Figure 3. Asymmetric elliptic jerk profile: jerk, acceleration, velocity, and position profiles for t _ad,pa = 0.3, t _ad,na = 0.5, t _ad,papj = 0.05, t _ad,panj = 0.15, t _ad,nanj = 0.2, t _ad,napj = 0.25.

The maximum absolute values of dimensionless jerk, acceleration, and velocity (208.07, 8.17, 1.634) correspond to the coefficients of jerk, acceleration, and velocity, defined as follows [Reference Biagiotti and Melchiorri7]:

(17) \begin{equation} C_{\mathrm{j}}=\frac{\max \!\left(\left| j\right| \right)}{h/T^{3}}, C_{a}=\frac{\max \!\left(\left| a\right| \right)}{h/T^{2}}, C_{\mathrm{v}}=\frac{\max \!\left(\left| v\right| \right)}{h/T} \end{equation}

The jerk profile of Fig. 3 is completely asymmetric, since all the phases have different durations, and all the four jerk peaks are different. This profile is named asymmetric elliptic jerk profile. The maximum absolute values of acceleration and jerk are lower in deceleration (phases #5−#7) with respect to acceleration (phases #1−#3), as usual in motion control to reduce residual vibrations.

The condition of lower absolute values of acceleration and jerk in deceleration can be maintained even with a simplifying assumption, which reduces the number of profile parameters. With the hypothesis that t _ad,papj = t _ad,panj and t _ad,nanj = t _ad,napj, from Eqs. (10) and (12), it follows that j _ad1 = j _ad3 and j _ad5 = j _ad7. Consequently, both the acceleration motion (phases #1–#3) and the deceleration motion (phases #5-#7) have symmetric acceleration profile, but can have different maximum values of acceleration and jerk. This profile is named symmetric elliptic jerk profile. In this case, the minimum set of parameters that define the motion profile is lower: four independent duration parameters (t _ad,pa, t _ad,papj = t _ad,panj, t _ad,na, t _ad,nanj = t _ad,napj) and two dependent peak parameters (j _ad1 = j _ad3 and j _ad5 = j _ad7). An example of symmetric jerk profile is shown in Fig. 4, which represents the dimensionless jerk, acceleration, velocity, and displacement for t _ad,pa = 0.3, t _ad,na = 0.5, t _ad,papj = t _ad,panj = 0.1, and t _ad,nanj = t _ad,napj = 0.2. The corresponding jerk peak parameters are j _ad1 = j _ad3 = 106.10, and j _ad5 = j _ad7 = 35.37. The dimensionless acceleration varies between a maximum positive of 8.33 in phase #2 and negative minimum of −5.56 in phase #6, while the dimensionless velocity reaches a maximum value of 1.667 in phase #4.

Figure 4. Symmetric elliptic jerk profile: jerk, acceleration, velocity, and position profiles for t _ad,pa = 0.3, t _ad,na = 0.5, t _ad,papj = t _ad,panj = 0.1, t _ad,nanj = t _ad,napj = 0.2.

4. Kinematic comparison with other motion laws

In this section, the proposed elliptical jerk motion profile (EJ) is compared from a purely kinematic point of view to other geometric and trigonometric motion laws analyzed in the scientific literature, maintaining the dimensionless approach for the sake of generality. The following motion profiles are compared in Fig. 5:

• TV: trapezoidal velocity profile [Reference Biagiotti and Melchiorri7]

• TA: trapezoidal acceleration (S-curve) [Reference Biagiotti and Melchiorri7]

• CY: cycloidal profile [Reference Biagiotti and Melchiorri7]

• SJ: sinusoidal jerk profile [Reference Valente, Baraldo and Carpanzano16]

• MSJ: modified sinusoidal jerk profile [Reference Fang, Qi and Hu17]

Figure 5. Comparison of the elliptical jerk law (EJ, red) with other motion laws: trapezoidal velocity (TV, blue), trapezoidal acceleration (TA, green), cycloidal (CY, black), sinusoidal jerk (SJ, cyan), modified sinusoidal jerk (MSJ, black dashed).

The cycloidal law cannot have finite phases with null jerk. Therefore, in order to include in the comparison also the cycloidal law, for the EJ profile a null duration has been imposed for the phases with null jerk (#2, #4, #6), along with an equal duration of 1/4 for the remaining ones (#1, #3, #5, #7). With these assumptions, all the considered motion profiles are fully defined, except the MSJ profile.

In general, the MSJ profile is characterized by 15 phases [Reference Fang, Qi and Hu17]. In the phases #1, #3, #5, #7, #9, #11, #13, and #15 jerk is sinusoidal; in the phases #2, #6, #10, and #14 the jerk is constant; in the phases #4, #8, and #12 the jerk is null. In practice, the MSJ profile is similar to the EJ profile, but each phase with semi-elliptical jerk is replaced by a triplet of phases (sinusoidal-constant-sinusoidal). To include in the comparison the CY law, the null jerk phases of the MSJ law (#4, #8, #12) have been eliminated. The constant jerk phases cannot be eliminated; otherwise, the MSJ law becomes a SJ law. Therefore, the following assumptions are made: the duration of the phases with sinusoidal jerk is 1/16, and the duration of the phases with constant jerk is 1/8.

Table I summarizes the coefficients of jerk, acceleration, and velocity (Eq. 17) for the considered motion profiles. The influence of these coefficients is extensively debated in ref. [Reference Biagiotti and Melchiorri7]. In general, smoother trajectories present higher peaks of velocity and acceleration (C _v and C _a). The coefficient C _a is significant since the force/torque applied by the motor is proportional to acceleration for inertial systems. On the other hand, the kinetic energy is proportional to square of velocity; therefore, it is convenient to have low values of C _v. The coefficient C _j provides an information about the jerk peaks, which influence the vibrations induced by a motion profile.

Table I. Coefficients of jerk, acceleration, velocity.

From the analysis of Fig. 5 and Table I, it is possible to summarize the following observations:

• As regards the coefficient of jerk, the SJ profile has the highest value (50.27), excluding the TV profile, which has infinite jerk, due to the acceleration discontinuities. The EJ profile has a coefficient C _j of 40.74, which is slightly higher than the ones of the CY and MSJ profiles (respectively 39.48 and 39.11). The TA profile has the lowest C _j (32.00), but has also jerk discontinuities for t _ad = 0, 0.25, 0.75, and 1. Jerk is discontinuous also for the CY profile in t _ad = 0, when it varies from 0 to 39.48, and in t _ad = 1, when it varies from 39.48 to 0.

• As regards the coefficient of acceleration, the EJ, TA, SJ, and MSJ profiles have the same value (8), while the CY and TV laws are characterized by lower values (2π and 4).

• With the considered hypotheses, the coefficient of velocity has the same values for all the profiles (2).

It is possible to note that the SJ and MSJ profiles are the most qualitatively similar to the proposed EJ profile as regards the jerk shape, but the SJ as a significantly higher C _j. On the other hand, the C _j of the EJ profile is very close to the ones of the CY and MSJ laws. A significant difference between the EJ profile and the TV, TA, and CY laws is that the EJ jerk profile is continuous.

In the following section, a dynamic comparison among these motion profiles is carried out with reference to a second-order linear system.

5. Dynamic comparison with other motion laws

For the dynamic comparison among the considered motion laws, a second-order linear system has been taken into account, characterized by the following differential equation:

(18) \begin{equation} m\ddot{x}=-k\!\left(x-x_{\mathrm{d}}\right)-d\!\left(\dot{x}-\dot{x}_{\mathrm{d}}\right) \end{equation}

This very simple linear model has been selected for the investigation since it can represent a wide variety of single-input single-output dynamic systems, such as:

• an inertia m (either translating or rotating), position-controlled by means of a PD closed-loop with proportional gain k, derivative gain d, and time-variable set-point x _d;

• an inertia m connected in parallel by a spring with stiffness k and by a damper with coefficient d to a moving base with time-variable position x _d.

The following results are referred to a case study with m = 1 kg and k = 2200N/m, considering three different levels of damping ratio ζ: (I) ζ = 0.1 (d = 9.38 Ns/m); (II) ζ = 0.5 (d = 46.90 Ns/m); (III) ζ = 1 (d = 93.80 Ns/m).

Six different time histories of x _d have been obtained scaling the six dimensionless profiles of Fig. 5 with a displacement h = 0.1 m and a motion duration T = 0.5 s. Figures 6–8 show the relative displacement x _r = x – x _d and the relative velocity dx _r/dt for the damping levels I, II, and III, comparing the vibrations deriving from the six motion profiles.

Moreover, Table II compares the six laws in terms of maximum absolute values of x _r, RMS values of x _r, settling times to within ± 0.04 mm (around 2% of the maximum relative displacement), maximum absolute values of relative velocity (|dx _r /dt|), and maximum absolute values of relative acceleration (|d ² x _r /dt ² |).

Figure 6. Dynamic comparison, case study I (ζ = 0.1): relative position x _r = x – x _d (top) and relative velocity dx _r /dt (bottom).

Figure 7. Dynamic comparison, case study II (ζ = 0.5): relative position x _r = x – x _d (top) and relative velocity dx _r /dt (bottom).

Figure 8. Dynamic comparison, case study III (ζ = 1): relative position x _r = x – x _d (top) and relative velocity dx _r/dt (bottom).

Table II. Comparison of the vibrations induced by the six considered motion laws.

From the analysis of Figs. 6–8 and of Table II, it is possible to outline the following conclusions:

• As expected, the TV profile, characterized by acceleration discontinuities and infinite jerk, induces the highest peaks of relative acceleration and velocity, and consequently the highest settling times, for all the damping levels.

• Also, the CY profile has high settling time with respect to the EJ, TA, SJ, and MSJ laws, especially in case I, with low damping; on the other hand, this profile induces the lowest peaks of relative acceleration and velocity.

• From a qualitative point of view, the dynamic responses induced by the EJ law are similar to the ones induced by the MSJ profile, for all the damping levels: as a matter of fact, in Figs. 6–8 the time histories of EJ and MSJ are nearly coincident; also the behavior related to the SJ profile is quite similar, but slightly more oscillatory.

• For all the damping levels, the maximum absolute relative displacement |x _r | of the EJ law is lower than the ones of the SJ and MSJ laws, but it is higher than the ones of the CY, TA, and TV profiles.

• The settling time of the EJ law is the second lowest in case I (lower damping), after the TA profile. In cases II and III, with higher damping, the settling times are more aligned for all the profiles, and the EJ profile is better than the TA profile. Summing the settling times in the three cases, the EJ profile is a second best after the TA law.