Effects of the weighting matrix on dynamic manipulability of robots
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Abstract
Dynamic manipulability of robots is a well-known tool to analyze, measure and predict a robot’s performance in executing different tasks. This tool provides a graphical representation and a set of metrics as outcomes of a mapping from joint torques to the acceleration space of any point of interest of a robot such as the end-effector or the center of mass. In this paper, we show that the weighting matrix, which is included in the aforementioned mapping, plays a crucial role in the results of the dynamic manipulability analysis. Therefore, finding proper values for this matrix is the key to achieve reliable results. This paper studies the importance of the weighting matrix for dynamic manipulability of robots, which is overlooked in the literature, and suggests two physically meaningful choices for that matrix. We also explain three different metrics, which can be extracted from the graphical representations (i.e. ellipsoids) of the dynamic manipulability analysis. The application of these metrics in measuring a robot’s physical ability to accelerate its end-effector in various desired directions is discussed via two illustrative examples.
Keywords
Manipulability Dynamic manipulability Operational space1 Introduction
To build a high performance robot, design is probably the most important process which hugely influences the robot’s performance. Designing a robot (i.e. determining the values of its design parameters such as mass and inertia distributions, dimensions, etc.) presets the limits of its abilities or in other words, its capabilities to perform certain tasks. If a robot is not well designed, no matter how advanced its controller is, it could end up in poor performance (Leavitt et al. 2004). In the other hand, if the design is “perfect”, the larger range of feasible options would be available in the control space which makes it easier for the controller to achieve a desired task with higher performance. Also, in case of redundant robots, a certain task is achievable via various configurations in which physical abilities of the robot are different (Ajoudani et al. 2017). Therefore, in order to improve the robot’s performance in different tasks and exploit its maximum abilities, it is desired to be able to compare different configurations of a robot and possibly to find the optimal one (e.g. in terms of torque/energy efficiency). This is completely intuitive since humans always try to exploit the redundancy in their limbs and also the environmental contacts to improve their performance while minimizing their efforts in executing various tasks. For example, usual human arms configurations are different while using screwdriver to tighten up a screw compared to while holding a mug.
As already mentioned, finding (i) proper values for the design parameters, and (ii) best configuration for a robot in performing a certain task are the two important elements in making high performance robots and/or improving the performance of existing robots. Thus, it is beneficial to develop a unified and general metric which enables us to measure physical abilities of various robots in different configurations and different contact conditions. For this application, there exists a very famous metric in the robotics community which is called manipulability. The concept of manipulability for robots first introduced by Yoshikawa (1985a) in the 80’s. He defined manipulability ellipsoid as the result of mapping Euclidean norm of joint velocities (i.e. \({\dot{\mathbf {q}}}^T {\dot{\mathbf {q}}}\)) to the end-effector velocity space. By using task space Jacobian (i.e. \(\mathbf {J}\)), he also proposed a manipulability metric for robots as \(w = \sqrt{\mathrm {det}(\mathbf {J}\mathbf {J}^T)}\) which represents the volume of the corresponding manipulability ellipsoid. The main issue with this measure is that, multiplying \(\mathbf {J}\), which is a velocity mapping function, and \(\mathbf {J}^T\), which is a force mapping function, is physically meaningless. In other words, in a general case, a robot may have different joint types (e.g. revolute and prismatic) and therefore different velocity and force units in the joints which makes the Jacobian to have columns with different units. This issue was first identified by Doty et al. (1995). They proposed using a weighting matrix in order to unify the units. However, even after that, many researchers used (Chiu 1987; Gravagne and Walker 2001; Guilamo et al. 2006; Jacquier-Bret et al. 2012; Lee 1989, 1997; Leven and Hutchinson 2003; Melchiorri 1993; Vahrenkamp et al. 2012; Valsamos and Aspragathos 2009) or suggested (Chiacchio et al. 1991; Koeppe and Yoshikawa 1997) same problematic metric for the manipulability of robots.
Yoshikawa (1985b) also introduced dynamic manipulability metric and dynamic manipulability ellipsoid as extensions to his previous works on robot manipulability. He defined dynamic manipulability metric as \(w_d = \sqrt{\mathrm {det}[\mathbf {J}(\mathbf {M}^T \mathbf {M})^{-1} \mathbf {J}^T]}\), where \(\mathbf {M}\) is the joint-space inertia matrix, and dynamic manipulability ellipsoid as a result of mapping unit norm of joint torques to the operational acceleration space. Here, \((\mathbf {M}^T \mathbf {M})^{-1}\) can be regarded as a weighting matrix which obviously solves the main issue with the first manipulability metric. However, physical interpretation of this metric still remains unclear. In other words, it is not quite obvious what the relationship is between \(w_d\) and feasible or achievable operational space accelerations due to actual torque limits in the joints. Although, Yoshikawa (1985b) and later on some other researchers (Chiacchio 2000; Kurazume and Hasegawa 2006; Rosenstein and Grupen 2002; Tanaka et al. 2006; Yamamoto and Yun 1999) tried to include the effects of maximum joint torques into dynamic manipulability metric by normalizing the joint torques, their proposed normalizations are not done properly and therefore the results do not represent physical abilities of a robot in producing operational space accelerations. The issue with their suggested normalization will be discussed in more details in Sect. 3.
Over the last two or three decades, many studies have been done on robot manipulability. Also many researchers have used manipulability metrics/ellipsoids in order to design more efficient robots or find better and more efficient configurations for robots to perform certain tasks (Ajoudani et al. 2015; Bagheri et al. 2015; Bowling and Khatib 2005; Guilamo et al. 2006; Kashiri and Tsagarakis 2015; Tanaka et al. 2006; Tonneau et al. 2014, 2016; Zhang et al. 2013). However, almost all of these studies have overlooked the effects of not using (or using inappropriate) a weighting matrix. In this paper, we focus on the weighting matrix for dynamic manipulability calculations and study its importance and influences on the dynamic manipulability analysis. We also show that, by using this analysis, we can decompose the effects of the gravity and robot’s velocity from the effects of robot’s configuration and inertial parameters on the acceleration of a point of interest (i.e. operational space acceleration). Therefore, the outcome of the dynamic manipulability analysis will be a configuration based (i.e. velocity independent) metric/ellipsoid which is dependent only on the physical properties of a robot and its configuration. Hence, we claim that, by selecting proper values for the weighting matrix, dynamic manipulability can provide a powerful tool to analyse and measure a robot’s physical abilities to perform a task.
This paper is an extended and generalized version of our previous study on dynamic manipulability of the center of mass (CoM) (Azad et al. 2017). Main contributions over our previous work are (i) generalizing the idea of weighting matrix for dynamic manipulability to any point of interest (not only the CoM), (ii) investigating the relationship between the dynamic manipulability and the Gauss’ principle of least constraints by suggesting a proper weighting matrix, (iii) describing the relationship between the dynamic manipulability metrics and operational space control, and (iv) discussing the applications of the dynamic manipulability metrics based on the suggested choices of weighting matrices.
We first derive dynamic manipulability equations for the operational space of a robot. To this aim, we use general motion equations in which the robot is assumed to have floating base with multiple contacts with the environment. Thus, the effects of under-actuation due to the floating base and kinematic constraints due to the contacts will be included in the calculations. As a result of our dynamic manipulability analysis, we obtain an ellipsoid which graphically shows the operational space accelerations due to the weighted unit norm of torques at the actuated joints. This is applicable to all types of robot manipulators as well as legged (floating base) robots with different contact conditions. The setting of the weights is up to the user which is supposed to be done based on the application. Two physically meaningful choices for the weights are introduced in this paper and their physical interpretations are discussed. We also discuss different manipulability metrics which can be computed using the equation of the manipulability ellipsoid. We investigate the application of those metrics in comparing various robot configurations and finding an optimal one in terms of the physical abilities of the robot to achieve a desired task.
2 Dynamic manipulability
The inequality in (17) defines an ellipsoid in the operational acceleration space which is called dynamic manipulability ellipsoid. The center of this ellipsoid is at \(\ddot{\mathbf {p}}_{vg}\) and its size and shape are determined by eigenvectors and eigenvalues of matrix \(\mathbf {J}_p \mathbf {W}_\tau ^{-1} \mathbf {J}_p^T\). As it can be seen, This matrix is a function of the weighting matrix \(\mathbf {W}_\tau \) and also \(\mathbf {J}_p\) which is dependent on the robot’s configuration and inertial parameters. Due to high influence of the weighting matrix on the dynamic manipulability ellipsoid, it is quite important to define \(\mathbf {W}_\tau \) properly in order to obtain a correct and physically meaningful mapping from the bounded joint torques to the operational space acceleration. This can be helpful in order to study the effects of limited joint torques on the operational space accelerations. Note that, if the weighting matrix is not defined properly, the outcome ellipsoid will be confusing and ambiguous rather than beneficial and useful.
3 Weighting matrix
In this section, we study the effects of the weighting matrix on the dynamic manipulability ellipsoid and propose two reasonable and physically meaningful choices for this matrix. First one is called bounded joint torques and incorporates saturation limits at the joints, and the second one is called bounded joint accelerations which assumes limits on the joint accelerations. The latter is also related to the Gauss’ principle of least constraints which will be discussed further in this section.
3.1 First choice: bounded joint torques
The shaded polygons in Fig. 1 represent exact areas in the acceleration space of the point of interest (i.e. the center point of the middle link) which are accessible due to the limited torques at the joints in six different configurations. These areas are computed using (11), numerically. As it can be seen in the plots, the polygons are always completely enclosed in the ellipses which implies that the dynamic manipulability ellipses, with the suggested weighting matrix in (18), are reasonable approximations of the exact feasible areas. These ellipses also graphically show that, given the limits at the joint torques, what accelerations are feasible in the operational space and what directions are easier to accelerate the point of interest. Note that, the choice of this point is dependent on the desired task. For example, for a balancing task, the CoM can be considered as the point of interest (Azad et al. 2017), whereas for a manipulation task, it makes more sense to choose the end-effector as the point of interest.
It is worth mentioning that, the main purpose of the plots in Fig. 1 is to show the accuracy of the approximation of the polygons by the ellipses. Although, one can compare the robot configurations in terms of feasible operational space accelerations with same amount of available torques at the joints. As it can be seen in this figure, the ellipses (and also polygons) in the left column are larger than their corresponding ones in the right column which implies that by changing the angle from \(90^\circ \) to \(120^\circ \), the range of available accelerations at the point of interest is extended.
3.2 Second choice: bounded joint accelerations
3.3 Relation to the Gauss’ principle of least constraints
Figure 2 repeats the graphs in Fig. 1 including new colored ellipses and areas. The blue, yellow and red ellipses show dynamic manipulability ellipses which are calculated using (27), where the joint weighting matrix \(\mathbf {W}_q\) is set to \(\mathbf {M}\), \(\frac{1}{4}\mathbf {M}\) and \(\frac{1}{9}\mathbf {M}\), respectively. Note that, the factor of \(\mathbf {M}\) in \(\mathbf {W}_q\) actually determines the norm of the inequality in (30). Obviously, this norm is 1, 2 and 3 for the blue, yellow and red ellipses, respectively. The colored polygons in the plots represent the corresponding exact feasible areas which are the results of mapping the joint accelerations in (30) to the task acceleration space given the torque saturation limits. These areas are obtained by evaluating (11) numerically subject to the inequality in the left hand side of (30) and also the torque limits.
The intersection areas between the colored ellipses and the black ones are shown in Fig. 3. The colored polygons in this figure are the same as those in Fig. 2. According to Fig. 3, the intersection areas are reasonable approximations of the exact areas shown by corresponding colored polygons. However, in the top two plots, the approximations are not as good as the other ones. The reason is that in these two plots, there are relatively large gaps between the feasible areas due to the torque limits only (i.e. gray polygons) and the dynamic manipulability ellipse with bounded joint torques (i.e. black ellipse) which directly affects the estimation of the colored areas. This is inevitable in some configurations for robots with under-actuation and/or kinematic constraints due to the rank deficiency of \(\mathbf {J}_q\).
As it can be seen in Fig. 2, the colored ellipses for each configuration have the same shape but different sizes. The shapes are the same since they are mapping the same equation (30), and the sizes are different since the values of the norm in this equation are different. The axis of the larger radius of the colored ellipses shows the direction in the task acceleration space in which lower inertia-weighted norm of \(({\ddot{\mathbf {q}}}-{\ddot{\mathbf {q}}}_u)\) is achievable. Hence, it is ideal to have the larger radii of both black and colored ellipses in a same direction to provide larger intersection area between them. In that case, larger part of the feasible area (i.e. the gray area which is estimated by black ellipse) would be covered by the colored areas implying that more points in the operational acceleration space will be achievable by lower inertia-weighted norm of \(({\ddot{\mathbf {q}}} - {\ddot{\mathbf {q}}}_u)\). In other words, although it is beneficial to have larger ellipsoids of both types (i.e. bounded joint torques and bounded joint accelerations with \(\mathbf {W}_q = \mathbf {M}\)), it is also desirable to have both ellipsoids in a same direction to maximize the intersection area between them.
4 Manipulability metrics
5 Applications of manipulability metrics
In this section, we explain the application of manipulability metrics through two examples. In these examples, we (i) compare different robot configurations (in Sect. 5.1), and (ii) find an optimal configuration (in Sect. 5.2) for a robot to accelerate its end-effector in desired directions. To this aim, the proper metric is the length of manipulability ellipsoid which is d in (33). The robot is assumed to be a three degrees of freedom RRR planar robot. Each link of this robot has unit mass and unit length with its CoM at the middle point.
5.1 Example I: Comparing robot configurations
Norm of the minimum joint torques and (black) ellipse lengths for six different robot configurations and three desired directions according to Fig. 5
Plot number | \(\rightarrow \) | \(\nearrow \) | \(\uparrow \) | |||
---|---|---|---|---|---|---|
\(||\varvec{\tau }||\) | \(d_1\) | \(||\varvec{\tau }||\) | \(d_1\) | \(||\varvec{\tau }||\) | \(d_1\) | |
1 | 0.20 | 4.43 | 1.22 | 0.70 | 1.77 | 0.49 |
2 | 0.32 | 2.73 | 0.59 | 1.44 | 0.92 | 0.94 |
3 | 1.18 | 0.73 | 0.18 | 4.80 | 1.18 | 0.74 |
4 | 0.69 | 1.26 | 0.22 | 3.85 | 0.83 | 1.04 |
5 | 2.14 | 0.41 | 1.50 | 0.57 | 0.19 | 4.65 |
6 | 1.02 | 0.85 | 0.67 | 1.28 | 0.27 | 3.20 |
Inverse inertia-weighted norm of the minimum joint torques and (gray) ellipse lengths for six different robot configurations and three desired directions according to Fig. 5
Plot number | \(\rightarrow \) | \(\nearrow \) | \(\uparrow \) | |||
---|---|---|---|---|---|---|
\(||\varvec{\tau }||_{\mathbf {M}^{-1}}\) | \(d_2\) | \(||\varvec{\tau }||_{\mathbf {M}^{-1}}\) | \(d_2\) | \(||\varvec{\tau }||_{\mathbf {M}^{-1}}\) | \(d_2\) | |
1 | 0.54 | 1.84 | 1.10 | 0.90 | 1.52 | 0.66 |
2 | 0.78 | 1.28 | 0.62 | 1.61 | 1.04 | 0.97 |
3 | 1.50 | 0.67 | 0.53 | 1.85 | 1.50 | 0.66 |
4 | 0.80 | 1.24 | 0.89 | 1.11 | 1.69 | 0.59 |
5 | 1.73 | 0.58 | 1.21 | 0.82 | 0.55 | 1.81 |
6 | 1.14 | 0.88 | 0.64 | 1.55 | 0.79 | 1.26 |
As it can be seen in both Tables 1 and 2, wherever the norm or the weighted norm of joint torques is higher the corresponding manipulability metric is lower and vice versa. In other words, norms or weighted norms of the torques are inversely related to the corresponding manipulability metrics \(d_1\) or \(d_2\). It implies that, maximizing manipulability metrics is the dual problem of minimizing the (weighted) norm of the joint torques. Therefore, one can optimize the relevant dynamic manipulability metric in order to maximize the robot performance or efficiency to perform a certain task. This will be described in the next example.
Another advantage of using dynamic manipulability analysis is that it provides a graphical representation of the mapping from the joint torques to the operational acceleration space which can help in better understanding the problem specially if it is a planar one. For example, comparing the plots in each row of Fig. 5, one can conclude that the left hand side ones are referring to better (more efficient) configurations for accelerating the robot’s end-effector in the desired directions. This is because both black and gray ellipses in the left column plots (odd numbers) are extended in the same direction as the desired ones, whereas in the right column plots (even numbers) at least one of the ellipses is not extended in the desired direction. This conclusion agrees with the values mentioned in the diagonal components of Tables 1 and 2 since the norm or weighted norm of the joint torques are lower in odd number plots compared to the corresponding even ones.
5.2 Example II: Optimizing the robot configuration
Norm and weighted norm of the minimum joint torques and lengths of the ellipses for two optimal robot configurations in Fig. 6
Plots | \(d_1\) | \(d_2\) | \(||\varvec{\tau }||\) | \(||\varvec{\tau }||_{\mathbf {M}^{-1}}\) |
---|---|---|---|---|
Top | 1.80 | 1.78 | 0.48 | 0.32 |
Bottom | 3.18 | 1.56 | 0.27 | 0.63 |
6 Conclusion
We revisited the concept of dynamic manipulability analysis for robots and derived the corresponding equations for floating base robots with multiple contacts with the environment. The outcomes of this analysis are a manipulability ellipsoid which is dependent on a weighting matrix, and different manipulability metrics which are extracted from the ellipsoid. We described the importance of the weighting matrix which is included in the equations and claimed that, by using proper weighting matrix, dynamic manipulability can be a useful tool in order to study, analyse and measure physical abilities of robots in different tasks. We suggested two physically meaningful options for the weighting matrix and explained their applications in comparing different robot configurations and finding an optimal one using two illustrative examples. The dynamic manipulability analysis can be performed for any point of interest of a robot according to the desired task.
Notes
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