Kinematic analysis of a rolling tensegrity structure with spatially curved members

The tensegrity simplex shown in Fig. 1a represents the simplest three-dimensional tensegrity structure. This structure consists of three struts, which are connected by 9 cables. The state depicted in Fig. 1a is a stable equilibrium configuration. With regard to the load in this configuration the struts are compressed members and the cables are classified as tensioned members. The replacement of the straight struts of the simplex by spatially precurved members yields a cylindrically shaped tensegrity structure as shown in Fig. 1b. The connectivity of the members is modified in a small manner regarding to original tensegrity simplex. Due to the spatial precurvature of the struts, there is a multidimensional stress state. Nevertheless, following the common terms in the field of tensegrity structures, these precurved struts are declared as compressed members.

To verify the occurrence of a free-standing self-stable equilibrium configuration a demonstrator of the varied tensegrity simplex is developed. This demonstrator is depicted in Fig. 1c. The precurved compressed members are made of aluminum and the tensionend members are realized by steel wire. Indeed, a stable equilibrium state occurs and the shape is described by cylinder with radius \(R=0.040\,{\mathrm {m}}\) and height \(H=0.150\,{\mathrm {m}}\).

Obviously, a cylinder features only a uniaxial bidirectional rolling locomotion. Thus, in order to realize a two-dimensional rolling locomotion a shape change of the tensegrity structure is required. This issue is realized by varying the length of selected tensioned members. Due to this actuation, the cylindrical geometry can be influenced to a conical shape which yields a circular rolling locomotion. The radius of curvature of the resulting trajectory depends on the conicity of the geometry. Finally, the combination of the rectilinear locomotion and the circular locomotion allows a general two-dimensional locomotion based on pure rolling. The actuation principle of the tensegrity structure to control the outer shape is shown in Fig. 2. The drive to realize the rolling of the system can be realized by the movement of internal masses as presented in [13].

The shape of the compressed members (\(j=1,2,3\)) is defined by the vector \(\varvec{\rho }_P\) parameterized respective to \(\varphi \) (\(\varphi \in [0,\varphi _{\mathrm {max}}]\) with \(\varphi _{\mathrm {max}}=4/3\pi \)). This issue is formulated in (1). The parameter \(\beta \) represents the cone angle of the structure. In this work, the complex deformation of the spirals is simplified as shown in (1) respective to the body fixed Cartesian coordinate system \(\{\xi ,\eta ,\zeta ,C\}\) (C represents the geometric center of the tensegrity structure). Hence, the height H of the cylinder and the cone is identical. Due to the limited shape change the parameter \(\beta \) can be varied within \(\beta \in \left( -\arctan \left( \frac{2R}{H}\right) ,\arctan \left( \frac{2R}{H}\right) \right) \). Applying the parameters of the demonstrator in Fig. 1c, the parameter \(\beta \) is limited approximately to \(\pm 28^\circ \).This issue clarifies that the radius of the cone in the \(\eta {-}\zeta \)-plane (\(\varphi =2/3\pi \)) is constant and cannot be influenced by the parameter \(\beta \).

The rolling locomotion system is represented by the considered tensegrity structure which is in contact with a horizontal rigid plane due to gravity as shown in Fig. 3. The position of the geometric center C of the structure is described by the vector \(\varvec{r}_C=(x_C,y_C,z_C)^{\mathrm {T}}\) respective to an inertial Cartesian coordinate system \(\{x,y,z,O\}\). The orientation of the tensegrity structure results due to three intrinsic rotations. This issue is given by a rotation respective to the z-axis of the inertial Cartesian coordinate system (rotation angle \(\gamma \)), a rotation respective to the resulting \(y'\)-axis (rotation angle \(\beta \)) and a rotation respective to the resulting \(x''\)-axis (rotation angle \(\alpha \)). To transfer the vectors defined in the body fixed coordinate system \(\{\xi ,\eta ,\zeta ,C\}\) into the inertial coordinate system \(\{x,y,z,O\}\) the transformation matrix \({\mathbf{T}} \) defined in (2) is applied.Thus, the vector \(\varvec{\rho }_P\) to the arbitrary body-fixed point P can be transformed into the inertial coordinate system as shown in (3). Here, the translational shift is taken into account using the vector \(\varvec{r}_C\).Hence, the current position of the locomotion system is characterized by the six parameters \(x_C\), \(y_C\), \(z_C\), \(\alpha \), \(\beta \) and \(\gamma \) which are summarized in the vector \(\varvec{q}\). However, the locomotion system enables only two degrees of freedom due to four nonholonomic constraints. In order to describe the behavior of the rolling locomotion system these constraints have to be taken into account. Therefore, the velocity of the tensegrity structure is considered. The velocity of an arbitrary body fixed point P is given in (4).The current contact point in the \(\eta {-}\zeta \)-plane of the tensegrity structure is denoted as Q and is specified by \(\varvec{\rho }_Q=(0,-R\sin (\alpha ),-R\cos (\alpha ))^{\mathrm {T}}\)). Additionally, two specific points attached to the tensegrity structure are considered. The point T represents the point of intersection of the \(\xi \)-axis and the \(x{-}y\)-plane. This point is given by \(\varvec{\rho }_T=(R/\tan (\beta ),0,0)^{\mathrm {T}}\). The parameter K represents a point on the circumference of the tensegrity structure in \(\eta {-}\zeta \)-plane characterized by \(\varvec{\rho }_K=(0,0,-R)^{\mathrm {T}}\). If K is in contact with the horizontal plane (\(\alpha =0\)) the corresponding velocity has to be zero. This issue yields three nonholonomic constraints given in (5). Additionally, the point T does not move perpendicular to the axis given by the current contact point Q and T (\(v_\bot =0\)). These issue allows the description of the fourth nonholonomic constraint which is given in (6).The parameter \(\dot{\alpha }\) and \(\dot{\beta }\) are chosen as generalized velocities. Considering the nonholonomic constraints in (5) and (6), the parameters \(\dot{x}_C\), \(\dot{y}_C\), \(\dot{z}_C\) and \(\dot{\gamma }\) can be formulated as shown in (7)–(10).The generalized velocities represent the actuation of the tensegrity structure. Here, \(\dot{\alpha }\) is realized by the movement of internal masses and the angular velocity \(\dot{\beta }\) represents the shape change of the structure. Following, the realization of the actuation of the locomotion system is not specified. Therefore, arbitrary actuation parameters \(\dot{\alpha }\) and \(\dot{\beta }\) are possible.

As first approach, the rolling locomotion of the cylindrical tensegrity structure without shape change is considered. Obviously, a rectilinear locomotion will occur. This issue is formulated by \(\beta _0=0\) and \(\dot{\beta }=0\) (\(\beta \equiv 0\)). Hence, the nonholonomic constraints formulated in (7)–(10) are simplified to (12)–(15).These constraints are integrable and are transformed into holonomic constraints by integration [18]. This approach yields a linear system of differential equation, which can be solved analytically. The resulting position is given by (16).Thus, this locomotion mode yields one degree of freedom specified by \(\alpha \). The simulation results depicted in Fig. 4 clarify the rectilinear locomotion. The orientation of the uniaxial trajectory is defined by the parameter \(\gamma _0\). Moreover, the direction of motion (forward locomotion/ backward locomotion) can be controlled by the actuation parameter \(\dot{\alpha }\).

In this section the locomotion behavior corresponding to the conical shape of the tensegrity structure characterized by \(\beta _0\) is considered. Apparently, a circular locomotion occurs due to the outer shape of the tensegrity structure. The corresponding nonholonomic constraints given in (7)–(10) are simplified by \(\dot{\beta }=0\) (\(\beta \equiv \beta _0\)). This approach is shown in (17)–(20).This locomotion enables only one degree of freedom specified by \(\alpha \). Moreover, the constraints listed in (17)–(20) are also integrable. Therefore, these constraints are integrated to holonomic constraints. The resulting linear system of differential equations is solved analytically. This yields (21).In order to clarify the circular locomotion of the conical shaped tensegrity structure, corresponding simulation results are depicted in Fig. 5.

These results confirm that the resulting locomotion trajectory is represented by a circle. The corresponding radius of curvature of the trajectory \(\rho _Q\) is given by \(|R/\sin (\beta _0)|\). Thus, the cone angle \(\beta \) enables the variation of \(\rho _Q\). The direction of motion (counterclockwise/clockwise) can be controlled by \(\dot{\alpha }\) or \(\beta \).

The previous locomotion modes are obvious and feature only one degree of freedom. For the realization of a general two-dimensional rolling locomotion system, the shape change of the tensegrity structure has to be taken into account. Hence, the nonholonomic constraints formulated in (7)–(10) cannot be simplified or be transformed into holonomic constraints. Due to the nonlinear character of (11) no analytical solution is known. Thus, the nonlinear system of equation given in (11) has to be solved numerically. The resulting rolling locomotion is simulated exemplarily for \(x_0=0\), \(y_0=0\), \(\beta _0=0\), \(\gamma _0=0\), \(\dot{\alpha }=\pi \,{\mathrm {rad/s}}\) and \(\dot{\beta }=0.4\ \cos (2\pi /5\ t)\,{\mathrm {rad/s}}\) for \(t\in [0\,{\mathrm {s}},10\,{\mathrm {s}}]\). The simulation results are depicted in Fig. 6.

With regard to the simulation results, indeed this locomotion type enables two degrees of freedom specified by \(\alpha \) and \(\beta \). The combination of the rectilinear locomotion corresponding to the cylindrical shape and the circular locomotion corresponding to the conical shape allows the realization of a general two-dimensional rolling locomotion without tilting sequences. In order to derive an appropriate locomotion strategy to control the locomotion behavior the resulting locomotion trajectories are considered in the following section.

The point Q represents the current contact point in the \(\eta {-}\zeta \)-plane of the tensegrity structure. The velocity of Q respective to the inertial coordinate system \(\{x,y,z,O\}\) is shown in (23).Thus, the trajectory of this point coincide with the horizontal plane. For the following investigations the actuation strategy of the rolling locomotion system is derived to ensure \(\varvec{\varGamma }=\varvec{r}_Q\). Therefore, these trajectories are evaluated as parametric equations. The properties of the trajectories characterized by the trajectory speed \(\dot{s}\) and the corresponding radius of curvature \(\rho \). Those characteristic parameters of the trajectory \(\varGamma \) are formulated in (24) and (25).

Analogous, the parameters are derived for the trajectory of the point Q. This characteristics are simplified applying (23) and (10). This issue yields the Eqs. (26) and (27).Obviously, the radius of curvature \(\rho _Q\) of this trajectory is limited respective to the possible shape change of the system. Therefore, a minimum radius of curvature given by the geometry of the tensegrity structure is defined: \(\rho _{Q,{\mathrm {min}}}=\sqrt{R^2+H^2/4}\). Thus, the trajectory \(\varvec{\varGamma }\) can be realized if \(|\rho _\varGamma |\ge \rho _{Q,{\mathrm {min}}}\ \forall t\). Finally, the actuation strategy is derived considering (28) and (29). Here, the sign of the parameter \(\dot{\alpha }\) is evaluated with regard to the tangent vector of the trajectory \(\varvec{\varGamma }\).These equations show that the actuation parameter \(\dot{\alpha }\) is responsible to guarantee to desired velocity of the trajectory \(\dot{s}_\varGamma \). The shape change of the tensegrity structure specified by \(\beta \) is required to follow the targeted path characterized by the radius of curvature \(\rho _\varGamma \). Finally, to define the actuation parameter \(\dot{\beta }\) the derivative of \(\beta \) respective to the time t has to be evaluated (\(\dot{\beta }={\mathrm {d}}/{\mathrm {d}}t\ \beta \)). This issue clarifies that the function \(\rho _\varGamma \) has to be differentiable to realize the desired trajectory by the rolling locomotion system. If discontinuities exist for \(\rho _\varGamma \) the locomotion system has to stop (\(\dot{\alpha }=0\)) to realize the corresponding radius of curvature. Therefore, the targeted path can be achieved. However, the velocity of the trajectory cannot be enabled.

Utilizing (28) and (29) the trajectory \(\varGamma \) can be realized for a suitable initial configuration \(\varvec{q}_0\) as shown in (30). Moreover, the radius of curvature of the trajectory \(\varGamma \) has to fulfill the criterion \(\rho _\varGamma \ge \rho _{Q,{\mathrm {min}}}\ \forall t\).

In order to highlight this advantageous and feasible actuation strategy the environment depicted in Fig. 7a is considered exemplarily. Because of the obstacles (grey rectangles) the targeted trajectory shown in Fig. 7a is defined by (31).An evaluation of the radius of curvature \(\rho _\varGamma \) confirms that \(|\rho _\varGamma |\ge \rho _{Q,{\mathrm {min}}}\ \forall t\). Thus, the realization of this trajectory is possible. The initial configuration is chosen considering (30). With regard to (28) and (29) the actuation parameters \(\dot{\alpha }\) and \(\dot{\beta }\) are calculated. These results are depicted in Fig. 7b, c. Finally, a numerical simulation of the locomotion system implementing the calculated actuation strategy confirms the locomotion on the targeted trajectory \(\varvec{\varGamma }\). Various states of the tensegrity structure during the rolling locomotion are depicted in Fig. 7a.

Thus, indeed the considered tensegrity structure with spatially precurved compressed members allows a steerable rolling locomotion on a horizontal plane due to the shape change between a cylindrical geometry and a conical geometry. Furthermore, the presented actuation strategy allows a feasible trajectory control of the rolling locomotion system to navigate through an arbitrarily shaped environment.