## I Introduction

Controlling the mechanisms of the machines and robots require an accurate mathematical model of the system. This model should contain all the physical characteristics of the system while it is computationally efficient. However, the control of the underactuated systems with passive bodies have certain challenges that can originate from the derived model [liu2013survey]. Physically, the underactuated systems [see Fig. 1] consist of two main parts: First, a rotating mass that moves by an actuator. Second, a passive body that displaces depending on the rotational mass.

Early studies for underactuated mechanisms have begun by the introduced Pendubot [spong1995pendubot] and Acrobat [spong1995swing] as the two-link manipulators. The general motion equations of a passive and an active rotating bodies with rotational angles of can be presented as

(1) |

where inertial matrix , velocity dependencies and gravity terms in and control inputs are defined by

The underactuated systems (1) with two degrees of freedom [liu2013survey] have a great common, similar inertial matrix , with certain underactuated spherical robots [kayacan2012modeling, svinin2015dynamic, TafrishiASME2019, TAfrishiRussiCha2019]. This inertial similarity help us to generalize our studying problem. The rolling spherical robots propel their passive carrier with a rotating mass-point [ilin2017dynamics, TafrishiASME2019, TAfrishiRussiCha2019] or pendulum [kayacan2012modeling, svinin2015dynamic] as Fig. 1.

From the control point-of-view, Agrawal in 1991 [agrawal1991inertia] found that when the inertial matrix of the inverse dynamics is singular at certain configurations, the integration of differential equations breaks. Arai and Tachi [arai1991position] showed that the inverse dynamics in a two-degree-of-freedom underactuated manipulator hit the singularity when coupled inertia term, the first constraint equation imposed from the passive body in (1), becomes zero and this property limits the domain of the control. Spong [spong1994partial] proposed a Strong Inertial Coupling condition for these underactuated systems under the positive definiteness of the inertia matrix. The condition grants a singular free inverse dynamics under certain geometric properties. In other words, these mathematical singularities that originate from the inversed terms of the inertial matrix , limit the mechanism to certain geometric parameters and create a challenge in manipulation around these singularity regions. Furthermore, a coupling index was proposed to determine the actuability of underactuated systems with different geometries [bergerman1995dynamic]. Later studies took place by following these coupling conditions [spong1994partial, bergerman1995dynamic] for controlling the Pendubot [zhang2003hybrid] and Acrobat [spong1995swing, spong1996energy]. The same problem was highlighted and control strategies are developed relative to this limitation for spherical robots [svinin2015dynamic, TAfrishiRussiCha2019]. Also, because the spherical carrier requires to have consecutive rotations without any angular limitations, the singularities due to inertial coupling become more challenging to deal with.

Our motivation in this letter is to introduce a new theory for modeling the underactuated systems, in which the limitations on the geometric parametrization and configuration singularities due to inertial coupling [spong1994partial, bergerman1995dynamic] are avoided. Thus, there wouldn’t be any physical design limitations due to mathematical singularities. This modified model is free from any complex algorithm. Additionally, the singularity regions in conventional rolling systems are analyzed for the first time. In this work, we begin by defining the combined phase-shifted curves that construct the circular rotation of mass. Next, the new kinematic model is applied to the Lagrangian equations for finding the rolling spherical carrier’s modified dynamics. Then, the inverse dynamics are derived and singularity regions are analytically demonstrated for this rolling system. Finally, a theory between combined waves and singularity regions is developed. This condition helps to design our combined waves proportional to the considered physical system. Furthermore, the theoretical findings are verified for a rolling carrier with an actuating mass-point by the simulations. Also, a classic model of the pendulum-actuated system is compared with our modified model to clarify the validity of the theory in simulation space.

For the rest of this paper, we show the kinematics of the combined phase-shifted curves with a rotating mass and also derive the non-linear dynamics in Section II. In Section III, by finding the inverse dynamics, the singularity-free conditions for the obtained model are explained. Finally, Section IV shows example simulations for a mass-point system with obtained singularity-free conditions and compare it with the classical model.

## Ii Modified Dynamical Model

In this section, we introduce sinusoidal trajectories that are combined around a circle. Next, the developed kinematics for a rotating mass is substituted into the Lagrangian function of a rolling system. Finally, the Lagrangian method is utilized to find the nonlinear dynamics of this underactuated model. The constructed trajectory with combined waves decomposes the terms of the non-linear model to sub-terms for each wave. Using this property, we will propose a theory that the singularities due to inertial coupling are removed through designing these combined waves.

### Ii-a Trajectory with Combined Sinusoidal Waves

Let us assume that the rotating mass has an orientation angle of with respect to the center of the spherical carrier with a radius of [see Fig. 2]. Also, the carrier is rolling with an angle of

. Then, the following position vector is defined by a sinusoidal curve on the circle with a radius of

as(2) |

where and are the amplitude of sinusoidal wave and the frequency of created periodic wave on the circle of radius , respectively. In the classic mass-rotating models, this trajectory becomes

(3) |

where and are zero in (2) that gives a circular rotation with radius [TafrishiASME2019, TAfrishiARM2019]. In our modified model, Eq. (2) is extended to sinusoidal waves where the mean creates the trajectory on circular reference path as

(4) |

where and are the frequency and phase shift of the th wave. Note that is a constant multiplier of wave’s frequency. Here, th circular wave has the phase shift where it is expected to combine number of these circular waves for having a constant circle with radius . We aim to design , and depending on the obtained relations from the inertial matrix to removes the coupling singularity ^{1}^{1}1Please check Theorem 1 for details. while rotating mass follows radius circle. Also, the deviation of trajectory with respect to circular radius can be found as

(5) |

where stands for the module of the complex variable. To have a circular rotation, we can easily prove from (5) that deviation always is assuming and are constant values for waves [see example for phase case with and in Fig. 2-a], so

(6) |

From the sine summations in (6), any number of the waves will always equal to zero. Thus, with considered assumption ( and ) for designing the waves, Eq. (4 ) becomes

(7) |

Note that as (6), hence, we will always have the circular trajectory with fixed radius (3) similar to the classic rotating mass systems. This proves that after deriving our general model by Eq. (4) and determining the required values for wave variables, combining these artificial waves will result a constant circular rotation with radius .

To find the rolling kinematics, the coordinate frames are sketched as Fig. 2-b. Here, represents the reference frame. The moving frame connected to the center of the spherical carrier is , which translates with respect to reference frame . Finally, is a rotating frame for the rotating mass-point attached to the center of spherical carrier and it is rotating with respect to . The corresponding kinematics for a rolling carrier with rotational mass is

(8) |

where , , and are the angular and linear velocities of the carrier and the angular and linear velocities of the rotating mass. Also, the linear velocity of the rotating mass is obtained by differentiating the Eq. (4) as

(9) |

### Ii-B Nonlinear Dynamics

The non-linear dynamics of the rolling spherical carrier with a planar motion is derived from the proposed trajectory equation. To find the corresponding motion equations, the Lagrangian equations are utilized.

We consider a sphere as a passive carrier (passive joint) where it is actuated with the rotation of a spherical mass as Fig. 2. The carrier has a mass of excluding the rotating mass. Also, the rotating mass with the mass is assumed as a mass-point. The Lagrangian function of the rolling carrier with the rotating mass, including kinetic and potential energies, along axis is described [TAfrishiARM2019] as follows

(10) |

where , , and

are the inertia tensor of rolling passive carrier, an arbitrary inertia tensor

connected to the mass-point, the acceleration of gravity and the distance of the mass-point respect to the ground, respectively. We include the inertia tensor for the sake of generality that its rotation is with the respect to carrier central frame . This arbitrary inertia tensor can be considered as either the lead of rotating pendulum [kayacan2012modeling, svinin2015dynamic] (yellow pendulum in Fig. 2) or interacting fluid/gas inside pipes for the rotating spherical mass [TAfrishiRussiCha2019] (blue fluid/gas in Fig. 2). After the substitution of Eqs. (8)-(9) into (10), one obtains(11) |

Finally, we apply the Lagrangian equations for planar translation along y axis as following

(12) |

where and are the external torques for the rotating mass and the sphere. Acting external torque between the surfaces of the spherical mass and carrier body is assumed zero, , since mass-point doesn’t contain any spinning around itself and it only rotates with the respect to the carrier center . After doing the necessary operations by Eqs. (11)-(12), the terms of the equations of the motion (1) for this underactuated system becomes

(13) |

while,

(14) |

Note that due to the linearity property of the summations, we can now interpret each wave () in Eq. (13) as a separate particle model in terms that the total average of these particles creates the dynamics of the rotating mass in a relation to the rolling carrier.

## Iii Inverse Dynamics and Singularity

In this section, the non-linear dynamics are presented in inverse form. A general condition for removing the singularity is derived and the singularity regions are analyzed for example systems. Next, we propose our theory for determining the parameters of the combined waves to avoid the singularity regions that originate from the coupled inertia matrix. Finally, a Beta function as feed-forward control for specifying the spherical carrier rotation is given.

The non-linear dynamics (1) with Eq. (13) are re-ordered with the goal to find the input torque from the specified rolling carrier states (). Hence, the rolling constraint of the carrier and the rotating mass differential equations in Eq. (1) becomes

(15) |

Now, we knew from (1) that inertial matrix is always a positive definite and symmetric matrix [agrawal1991inertia] where upper-left determinants grant this condition by and . To extend these conditions to the derived inverse dynamics, the rolling constraint (first differential equation) in Eq. (15) is substituted into the second differential equation as follows

(16) |

where

(17) |

Because the mass-point and the carrier rotation are opposite of each other in our motion and for the sake of the simplicity, we assume . By relying on the Ref.[spong1994partial], the coupled inertia matrix should be positive definite as well. However, denominator in requires another extra condition that . Thus, under the condition of , there exist singularities in the solution of Eq. (16) for the cases when () [agrawal1991inertia]. Thus, following proposition as the condition of the singularity is expressed.

###### Proposition 1

###### Proof

Before designing our combined-waves model under the Proposition 1, we check the singularity regions for the different classical underactuated rolling systems. Note that in these cases the trajectory is considered as an ideal circle (3) without any consideration of our combined sinusoidal curves.

###### Example 1

Singularity region of a classical rotating mass-point system [TAfrishiARM2019, TafrishiASME2019], where , are analyzed using condition (18) in Proposition 1. Let the trajectory be a perfect circle with radius , which makes and as (3) when . From the given (18) condition, the inequality is transformed to

(21) |

Now, by considering the maximum possible value for , one obtains a limitation on the geometric parametrization as

(22) |

This means in designing this mass-point system, the inverse dynamics model (15) will hit singularity if the radius of rotating mass be less than rolling carrier as condition (22) and singularity disobeys the physical mechanics completely. Fig. 3 shows how changes in the geometric parameters () in (21) affect singular configurations of the mass-point (3) on the steady spherical carrier (). This graphic clarifies that rolling systems without any angular constraint on and will hit the singularity. Otherwise, the solution of (15) will break many times while this singular region changes by spherical carrier rotation, .

###### Example 2

A rotating mass system with arbitrary inertial tensor is chosen in this example. This inertia tensor can be related to a rod that connects the mass to the center of the rolling body [kayacan2012modeling, svinin2015dynamic] or an interacted water with the rotating mass in pipes [TAfrishiRussiCha2019, TafrishiASME2019]. Thus, with a circular trajectory as the previous example, condition (22) is transformed to

(23) |

By sorting this condition based on the with considering , we see that singularity can be avoided only when

(24) |

Similar to the previous example, singularity limits the inverse dynamics for only certain mechanisms that can satisfy the following geometric condition.

To show that our proposed approach removes the demonstrated singularity regions in Eq. (21) and Eq. (23), and how parameters of the combined waves should be designed analytically a theory is developed.

###### Theorem 1

The trajectory with multiple combined sinusoidal waves never hit singularity and positive definiteness of is granted when variables and of the combined waves in this model are designed for with satisfying following inequalities

(25) |

where

###### Proof

Let the singularity condition (18) from Proposition 1 be

To have the right-hand side of the inequality always larger than the left-side, we find the absolute value of left-side in the three cases as

(26) |

where

. Next, we utilize the Fourier Transform equations

[vretblad2003fourier] as follows(27) |

where and are the transformed term of and the frequency of corresponding . With applying Fourier Transform (27) to each side of inequalities in (26) under linearity property [vretblad2003fourier], one obtains

(28) |

where is the Fourier Transform of the inertia tensor of , while the relevant terms are

(29) |

By using transformed equation (29), the combined waves are simplified to two base waves for comparison: the first term is the constant shift by and the second is the sinusoidal waves, . Because the angular rotation of the waves in both sides of inequality is always same, each side of (28) can be compared relative to its multiplier with the same frequency . As an interesting point, despite the usage of waves in different phases , they are simply canceled out from both sides of inequality (28) with the help of Fourier Transformation [vretblad2003fourier]. By the known insight in the expressed properties, all three conditions in (28) are collected for each specific impulse in the given frequency in th wave in the following form

(30) |

To keep the left-hand side always be larger than the right-hand side of inequalities, the moduli (presented like ) of these complex transforms (30) are calculated

(31) |

Because, the amplitudes of the combined waves doesn’t give clear comparison, the conditions are extended by the phase difference between each of the sides. Thus, by checking (30), we see that there are and phase differences between each side of sinusoidal curves in conditions 1 and 2, respectively. Then, the sinusoidal parts of the waves are represented from (30)-(31)

(32) |

By taking the Inverse Fourier Transform from (32) and calculating the required shifts of each comparison, the minimum shift requirements [see the Appendix for derivations], for a right-hand side larger than the left-hand, are calculated

(33) |

By substituting Eq. (31) and Eq. (33) back to inequality (28) and taking the Inverse Fourier for single wave , one obtains

(34) |

To simplify the second term at right-hand side of inequalities (34), we assume that always which results to transfer to in all conditions . Under the given assumption (), the Eq. (25) can be derived from (34) with including combined waves summation.

###### Remark 1

Because inertia tensor of the rotating mass is normally related to geometric objects (connecting cylindrical bar of pendulum) with a constant radius, it has been included as the constant value to the inequality.

###### Remark 2

This Theory 1 can easily be extended for any underactuated system with two-link manipulators (for example the Acrobat) since term is in common in all models and does not have the inertia tensor of carrier .

In this study, we choose a 4th order Beta function [svinin2015dynamic, TAfrishiRussiCha2019] to arrive the carrier toward its desired final configurations by

(35) |

where and are the time constant of designed motion and the value for the final arrived distance . We expect from this feed-forward control to actuate the rotating mass like from (35) as a two-step motion. This two-step motion of rotating mass [see at Fig. 6-b as an example of this motion pattern] is followed by a counterclockwise rotation till certain angle and a similar clockwise rotation for returning to the rest position. Note that similar to what has been developed in [svinin2015dynamic, TAfrishiRussiCha2019], one can show that with the selection of this motion scenario the condition is always satisfied.

## Iv Simulation Analysis

In this section, the proposed theory is analyzed in the simulation space. At first, to evaluate the condition in the worst-case scenario, a mass-point system with is chosen. We find the singular-free model with satisfying the proposed Theorem that uses the combined phase-shifted waves. Next, to show that the modified model is similar to the classic model, we compare both cases when there is an inertia tensor as a pendulum system.

Variable | Value | Variable | Value |
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