Relative Robots: Scaling Automated Assembly of Discrete Cellular Lattices

We propose metrics for evaluating the performance of robotically assembled discrete cellular lattice structures (referred to as digital materials) by defining a set of tools used to evaluate how the assembly system impacts the achievable performance objective of relative stiffness. We show that mass-specific stiffness can be described by the dependencies E∗(γ,D(n, f ,RA)), where E∗ is specific modulus, γ is lattice topology, and the allowable acceptance of the joint interface, D, is defined by an error budget analysis that incorporates the scale of the structure, and/or number of discrete components assembled, n, the type of robotic assembler, RA, and the static error contributions due to tolerance stack-up in the specified assembler structural loop, and the dynamic error limitations of the assembler operating at specified assembly rates, f . We refer to three primary physical robotic construction system topologies defined by the relationship between their configuration workspace, and the global configuration space: global robotic assembler (GR), mobile robotic assembler (MR), and relative robotic assemblers (RR), each exhibiting varying sensitivity to static, and dynamic error accumulation. Results of this analysis inform an iterative machine design process where final desired material performance is used to define robotic assembly system design parameters. INTRODUCTION Digital materials exhibit coded functionality by programmatically defining the type, and location of homogeneous or heterogeneous discrete building blocks such that their mechanical properties combine to perform as an explicitly defined continuum material. One example of a highly stiff, and ultra-light material is a high connectivity, non-stochastic, periodic, lattice structure composed of axially loaded truss elements [1]. Differentiating from 3D printing, discretization of the cellular lattice into reusable building-block elements enables fabrication, and reconfiguration, of explicitly defined heterogeneous meta-materials. The regular, periodic nature of the discrete cellular lattice can be exploited to simplify automated assembly by robotic processes. In this paper we lay out a methodology to identify the overall performance of robotic assembly of discrete cellular lattice with metrics based on machine class, scale of assembled material, and assembly rate. The critical dependency of robotic assembly is the ability for the interface between joined discrete cells (voxels) to accommodate error inherent to the assembly process while maintaining robust force/energy transfer across the node. Performance of discrete construction of three dimensional periodic lattice structures is based on the behavior of cellular solids with properties governed by their constituent material, and lattice topology [2]. Analogous to naturally occurring cellular materials such as bone, and sponge, these engineered periodic lattices act as continuum meta-materials [3], which can achieve ultralight stiffness to weight ratios by following relative density linear scaling relationships from the base material to the lattice [4, 5]. In discrete lattice construction, the parasitic mass contribution of the interface affects overall system mass-specific stiffness. Given a base material with youngs modulus Es, and density ρs, ideal stretch dominated behavior with specific modulus 1 Copyright c © 2016 by ASME E∗ at density ρ∗ follows a proportional law of E ∗ Es ∝ ρ ∗ ρs γ , where γ varies with lattice topology, and connectivity (Fig. 1) [3, 5]. In this paper, we will assume a single periodic cubic octahedra lattice topology with γ = 1.5. FIGURE 1. SPECIFIC MODULUS COMPARISON CHART. FIGURE 2. COORDINATE FRAMES OF MAJOR LINKS, and THEIR OFFSETS FOR AN HTM ANALYSIS OF THIS CUSTOM BUILT MR-TYPE ASSEMBLER. OL IS THE BASE FRAME ORIGIN (LOCATED ON THE LATTICE), OE IS THE TOOL FRAME. Robotic assembly of truss structures has typically taken the form of multi-degree of freedom (DOF) industrial robot arms that place discrete truss segments into larger structural configurations [6]. In order to increase attainable build volume, linear stages are built for the arm to translate around the structure it builds. Hoyt et al have developed an approach for on-orbit space assembly, using several multi-DOF arms to enable robot locomotion, truss manipulation, and assembly [7]. Terada et al have simplified the structure-robot system by designing the robot relative to the parts it places, unifying locomotion, and part placement [8]. We refer to three primary physical assembly system topologies that are defined by the relationship between their task-space, and global work-space coordinate frames: GR (global robotic assembler), MR (mobile robot), and RR (relative robot). Figure 2 shows a mobile robot, and Fig. 3 shows example assembly systems from each category. Static, and dynamic error is accumulated differently in each machine class. Assembler static error is composed of constant, systematic structural loop tolerances due to machine kinematics, and manufacturing. The dynamic response of the assembly system to self, and forced-excitation defines a dynamic error contribution based on system configuration. The speed at which parts can be placed, the build frequency (f), is a function of robot type (RR), and is defined by the interface geometry (D). The dynamics of the assembly robot contribute a vibrational error that decays toward a target position. The interface geometry accommodates the assembly system static, and dynamic error. The larger the allowable acceptance of the interface the shorter required settling time, resulting in increased allowable assembly rates. Static error also accumulates in the incremental assembly of the lattice due to accumulated tolerance stack-up error of the assembled lattice elements. Each machine class exhibits varying sensitivity to this lattice tolerance stack-up, and plays a part in system choice. The combination of static, and dynamic errors defines a maximum positional variation at the interface (D) of the assembly front, and a minimum geometric interface condition to accommodate the positional tolerance. Larger tolerance management requirements grow the interface geometry, increasing interface node mass. Tighter tolerance assembly allows decreasing node interface geometry, which decreases mass, thereby increasing mass-specific stiffness. We propose metrics for evaluating the performance of robotic assembly of discrete cellular lattices based on system error accumulation, mitigation of this by an allowable acceptance at the discrete interface, and those affects on material performance. METHODOLOGY We aim to define the dependencies of robotic assembly of discrete cellular lattice structures as a set of tools used to evaluate desired performance objectives. Specific stiffness, ultimately dependent on lattice topology (a factor that is a geometric constant), is affected by the mass of the mechanical interface between discrete voxels, as well as the mass of the lattice strut elements. 2 Copyright c © 2016 by ASME FIGURE 3. RELATIVE ROBOT CONFIGURATIONS (L TO R): GLOBAL (GR), MOBILE (MR), RELATIVE (RR). WHERE, a = GLOBAL COORDINATE FRAME, b = BASE FRAME, c = CONFIGURATION SPACE, d = WORK SPACE (SHADED REGIONS). Geometric limitations of the interface geometry of the discrete cellular lattices are quantified by the type of robotic assembler, the number of discrete components assembled, the static error contributions due to tolerance stack-up in the assembler structural loop, and the dynamic error limitations due to assembly rate. This relationship is summarized with the following dependency: E∗(γ,D(n, f ,RA)) (1) E∗ is the specific modulus of the assembled material, γ is a constant describing lattice topology, D is the interface geometry, n is number of parts composing the material (which can also be interpreted as scale of the overall geometry being built), f is part placement frequency,and RA refers to the type of robotic assembly system in use. Equation 1 states that for an assembled material with a desired specific modulus there exists a geometrical limit to the voxel interface. The dimensions of this interface are based on the relationship between desired operating frequency, chosen assembly system, and the number of elements to be placed by a specified type of assembly robot. Ultimately, material performance is a function of each of the metrics shown in Eqn. 1. Each variable in this equation will now be described in greater detail. Robotic Assembler Taxonomy (RA) The robotic assemblers are categorized into three distinct system types defined by the relationship between their base frame, the global coordinate frame of the final assembled structure, and the assemblers configuration space [9]. A static datum at the origin of the lattice is defined for all system topologies as the global coordinate frame. The extents of the lattice (n voxel elements) define a global configuration space in R3. Each robot class establishes a configuration(s) that can reach a subset of this configuration space defined by the kinematic limits of its workspace. The extents of the workspace are defined by the number of units of voxels, m, that can be reached by the tool frame in a single configuration. The base frame of the global robotic assembler is static. The mobile assembly robots, having a workspace that is a subset of the configuration space must move across the lattice during the assembly process in order to reach the extents of structure. For each configuration these mobile robots establish new base frames relative to the global coordinate frame. Global Robotic Assembler The global robotic assembler (GR) workspace reaches the full extents of the assembled structure (m = n). Since a single configuration is required for the tool frame to reach the entire workspace the base frame is aligned with the global coordinate frame. All motions of the end-effector tool frame are performed with respect to this static global reference [10]. We define such a machine as consisting of an analog motion system a machine that can perform continuous, and arbitrary motions (dependent on its positional tolerance, and the kinematic limits of its drive system configuration) across the entire range of the assembled structure. During the assembly process voxels are placed with reference to this origin datum at the global coordinate frame. Generally a machine of this type is not mass or stiffness limited, thereby enabling high precision endeffector placement. However, the already assembled structure becomes part of the overall structural loop as voxels must interface with already placed components. As the assembly front approaches the further extents of the machine, tolerance stack-up of assembled components contribute to positional uncertainty at the target-voxel interface. Mobile Robotic Assembler The mobile robotic assembler (MR) system is an assembly system that has similar kinematic configuration to a GR except the workspace is a subset of the total configuration space (m < n), and as such also includes a motion system to move across the lattice. In order to reach the extents of the configuration space the mobile assembler translates relative to the assembled lattice, establishing a new base coordinate frame after each locomotion maneuver to extend the 3 Copyright c © 2016 by ASME effective workspace of the tool frame. Similar to the GR, lattice elements contribute to the structural loop accumulating error. For the MR the structural loop consists of a reduced number of elements but includes additional structural components in the locomotion system. Relative Robotic Assembler The relative robotic assembler system (RR) has a workspace that encompasses a minimal unit step (m ' 1). The kinematics are tuned to the geometry of the lattice, limiting motions to discrete or digital motions rather than analog continuous motions. Translation across the lattice is made in unit steps defined by the kinematics of this assembler. As the robot traverses the lattice it establishes a base coordinate frame at each step, places or removes an element, and then traverses to an adjacent cell, referencing only the previous, adjacent cell. In this way the tool frame is also the incrementally established base frame. This mobile relative robot minimizes the accumulation of error by reducing the structural loop to only neighboring elements, enabling the reference to be relative to the robot rather than the structure. The cost of this topology is complexity in locomotion, and material handling (not addressed here). Scale (n) While not explicitly an objective variable, scale (overall dimension or number of elements) of the assembled structure is a discerning factor in choosing a robotic assembly system. Assuming isotropic assembly (assembly in any principal direction), for n number of voxels with strut length L, the volume of assembled structure will have uniform x, y, and z dimensions of: d = L √ 2 · 3 √ n (2) This scaling will become important as we evaluate the consequences on robot selection for building very large scale structures where the extent of the material continuum surpasses a physically realizable machine enclosure size, and where the tolerance stack-up of placed parts becomes a substantial consideration. Interface (D) The interface joints between discrete lattice parts is the energy transfer point of the overall material; the interface may transmit force, moments, power, data, etc. This requires the geometry of the interface to accommodate docking of gendered features. The allowable acceptance of these features must accommodate the translational, and rotational errors of the assembly process, and provide passive alignment compensation in addition to energy transfer [11–13]. For simplicity of the discussion we consider only two dimensional interfaces with a single degree of freedom translational error the approach direction is excluded FIGURE 4. DISCRETE INTERFACE GEOMETRY. A) JOINT VOLUME CONTRIBUTION, B) ALLOWABLE ACCEPTANCE DI-