# Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot
## 4 Elliptic Jerk Motion Profile  
Figure 3 shows the elliptic jerk motion profile [10] in dimensionless formulation, adopted for the sake of generality. For any motion displacement \(d\) and motion duration \(T\), it is possible to define:  
* the dimensionless time, which is the time \(t\) divided by \(T\): \(t_{\text{ad}}=t/T\);
* the dimensionless position, which is the position \(s\) divided by \(d\): \(s_{\text{ad}}=s/d\);
* the dimensionless velocity, which is the velocity \(v\) divided by \(d/T\): \(v_{\text{ad}}=vT/d\);
* the dimensionless acceleration, which is the acceleration \(a\) divided by \(d/T^{2}\): \(a_{\text{ad}}=aT^{2}/d\);
* the dimensionless jerk, which is the jerk \(j\) divided by \(d/T^{3}\): \(j_{\text{ad}}=jT^{3}/d\). The elliptic jerk profile is divided into seven phases (#1-#7, Figure 3). The \(i\)-th phase starts at \(t_{\text{ad,i-1}}\) and ends at \(t_{\text{ad,i}}\). Phases #1-#3 are characterized by positive acceleration, and their overall duration is \(t_{\text{ad,pa}}\). The fourth phase is characterized by constant velocity and null acceleration. Phases #5-#7 are characterized by negative acceleration, and their overall duration is \(t_{\text{ad,na}}\). Since the dimensionless duration of the motion is a unit, the duration of the fourth phase is \((1-t_{\text{ad,pa}}-t_{\text{ad,na}})\). Positive acceleration (phases #1-#3) is performed with a positive jerk in phase #1, with duration \(t_{\text{ad,paj}}\), with a null jerk in phase #2, and with a negative jerk in phase #3, with duration \(t_{\text{ad,paj}}\). Consequently, the duration of phase #2 is \(t_{\text{ad,pa}}-t_{\text{ad,paj}}-t_{\text{ad,paj}}\) (Figure 3). Similarly, deceleration (phases #5-#7) is performed with a negative jerk in phase #5, with duration \(t_{\text{ad,naj}}\), with a null jerk in phase #6, and with a positive jerk in phase #7, with duration \(t_{\text{ad,paj}}\). Consequently, the duration of phase #6 is \(t_{\text{ad,na}}-t_{\text{ad,naj}}-t_{\text{ad,naj}}\) (Figure 3). For the phases with a non-null jerk (#1, #3, #5, and #7), the jerk profile is assumed to be elliptic. The jerk peaks of these phases are, respectively, \(+j_{\text{ad1}}\), \(-j_{\text{ad3}}\), \(-j_{\text{ad5}}\), \(+j_{\text{ad7}}\). Cons  
Figure 3: Generic elliptic jerk profile. quently, the dimensionless jerk profile is completely defined by ten positive parameters: four jerk peak parameters (\(j_{\rm ad1,j}\), \(j_{\rm ad3}\), \(j_{\rm ad5}\), \(j_{\rm ad7}\)) and six time parameters (\(t_{\rm ad,pa}\), \(t_{\rm ad,na}\), \(t_{\rm ad,papi}\), \(t_{\rm ad,papi}\), \(t_{\rm ad,napi}\)). The acceleration, velocity and position profiles are obtainable by successive integrations, so they are defined by the same parameters. Nevertheless, these parameters are not independent, since the following four conditions hold for a generic dimensionless rest-to-rest motion: null acceleration at the end of phase #3, null acceleration at the end of phase #7, the null velocity at the end of phase #7, and unit dimensionless displacement at the end of phase #7. These four conditions can be used to obtain the four jerk peak parameters as a function of the six time parameters. The analytical expressions of the dimensionless jerk, acceleration, velocity and position, and the expressions of the jerk peak parameters as functions of the time parameters are discussed in detail in [4] and are not reported here for reasons of space. The number of independent dimensionless profile parameters can be further reduced by adding the following two hypotheses: \(t_{\rm ad,papi}=t_{\rm ad,papi}\), and \(t_{\rm ad,napi}=t_{\rm ad,napi}\). It is possible to demonstrate that these two further assumptions lead to symmetric peak profiles both in acceleration and in deceleration (\(j_{\rm ad1}=j_{\rm ad3}\) and \(j_{\rm ad5}=j_{\rm ad7}\)) [10].