The authors formulate their planning system using constrained optimization to solve for both the position of the base of their RRR robot and movement of the RR chain that achieves the desired task. I cannot help but wonder why they did not plan the trajectory of the end point in polar coordinates relative to the base joint, and use the extra degree of freedom to separately position the links of the manipulator in ways that achieve their obstacle avoidance constraints. I am certain that I am missing something. This is an interesting paper complete with the analysis necessary to understand it implementation. I recommend acceptance for publication.

The main interest of the proposed method is to avoid discretization of the path which may, as illustrated in the examples given in the paper, lead to feasibility issues. The video provided with the paper nicely shows the possibilities offered by this work. On overall, the paper is well organized and presented. However, in my opinion, the following concerns must be dealt with. 1. The main issue with the method is its applicability limited to planar paths, obstacles in form of circles and planar 2 and 3-DOF manipulators. These important limitations are not explained, not even mentioned, in the abstract, introduction and conclusion of the paper. The literature review, comparing this work with previous ones, should also account for these limitations by providing fair comparisons with previous methods which can be applied to spatial cases. 2. The main tool allowing the method to avoid discretization is the use of swept volume (SV) calculations. Regarding these calculations, I have the following concerns: 2a. There is not literature review on this topic in the paper which makes it difficult to assess the detailed technical contribution of the paper. Unless I miss something, he SV calculations mainly amounts to simple planar convex hull calculations which tends to show that the technical contribution is limited. 2b. The method is mainly limited to planar 2-DOF manipulators since the SV for 3-DOF manipulators is calculated assuming that only the last two links are moving during the execution of a shape primitive which, essentially, makes the 3-DOF manipulator having 2 DOFs. 2c. The extension of the method to spatial paths and manipulators with 3 or more degrees of freedom may be very challenging. This extension is mentioned in the future works but there is no clue of the difficulties linked to SV calculations in 3D and for n-DOF manipulators with n>3 (notably on the issue of multiple inverse kinematics solutions). 3. The choice of the optimization problem cost function as the sum of link lengths is not motivated. Several other criteria, as found in the literature, could have been used so why this one? Why not a multi-objective optimization problem formulation? 4. What about possible issues with kinematics singularities and what about the timing and synchronization of the manipulator joints along the path?

In this work the authors present a constrained optimization approach to computationally determining a design and trajectory for a multi-link (2 or 3) planar robot in order to enable it to follow a continuous, user specified end effector trajectory on a 2D plane in the presence of obstacles. The authors demonstrate the efficacy of their method on a fully implemented physical system that includes a user interface for detailing circular obstacles in the robot’s environment, constraints on the robot’s base position, and the continuous trajectory that must be followed by the end effector. The authors show that the method is successfully able to generate a design for 3 trajectories, and then show the robot performing the task in the physical world after fabricating the design produced by the method. Additionally, the authors favorably compare their method to one that does not consider continuous trajectory following but rather discretized trajectories based on a sampling-based motion planning method. The strengths of the paper include the impressive end-to-end demonstration of the method’s use from task definition to physical execution on a robot constructed according to the designs produced by the method. This step, physical construction of multiple designs, is one that is often missing in design optimization work and is appreciated here. A related strength is that the paper demonstrates a complete system, including the user interface for defining the task and constraints. Consideration of the interface through which users define tasks and constraints is an important aspect of building useful robotic systems that is frequently overlooked. Its inclusion here is appreciated. Additionally, for the most part the method is thoroughly described and the paper is generally well written. Further, the video attachment is valuable and demonstrates the results nicely. There are some concerns that would strengthen the paper were they addressed. In no particular order: 1) In some places in the paper, the descriptions of aspects of the method for the 2-link case are thorough, but the description of how that is extended to 3 links is not sufficiently detailed to be clear. For instance, in the last two paragraphs of Section IIIa, the SV simplification for the 3DOF case is done by not allowing the proximal link to move during the execution of a shape primitive. The implications of this simplification need to be described. Does this simplification have a meaningful effect on the method’s ability to find valid solutions? This needs described in more detail and the implications should be discussed. Similarly the following paragraph which describes the “changing origin” simplification should be expanded upon and the implications discussed. 2) In some cases the terminology is not conducive to clarity and may be inconsistent or utilized prior to being defined. For instance j, u, I, f, closest, and farthest in the constraints on equation 1. These can be inferred with some difficulty but should be explicitly defined prior to their use. 3) It is appreciated that the authors note that the method’s success is subject to its initialization. As the method is constructed as a very highly constrained optimization problem and implemented using an optimization algorithm that doesn’t provide global guarantees, this is potentially a major issue. This is partially addressed by the “Initial guess” paragraph in the implementation paragraph, however this issue is worthy of more thorough evaluation and discussion. How frequently does the optimization fail to find a solution? The ramifications of this issue would become more clear if quantitative evaluation of this issue was included in the work, perhaps via some notion of randomized trajectories and randomized environments. Such an evaluation would then in turn would help the reader understand the likelihood of success of the method with respect to desired trajectories outside of the three currently evaluated. 4) Building on the last point, inclusion of the computation time analysis in the paper is appreciated, but could be more thorough. Computational time required by constrained optimization is generally heavily dependent on the specific problem instance and as such the computational time required by the method isn’t sufficiently evaluated by only 3 problem instances. If randomized trajectories or environments were included as described in the previous point, the timing results would provide more insight into the time required by the method across more problems. Further, the RRT-based comparison method, as a sampling-based method, is going to be dependent on the random seed it is initialized with. It would be valuable to run the RRT-based method multiple times and report statistics on the timing results for this to be more meaningful. 5) The method is specifically tailored to end effector trajectories in R^2 to be followed by a planar manipulator of 2 or 3 links. However most of your motivating examples in the introduction would require R^3 or SE(3) end effector trajectories to be executed on more complex, non-planar manipulators. The paper would be strengthened if the case solved by the method (Planar trajectory with 2 or 3 link planar manipulator) is given more motivation. I do appreciate that an extension to 3D trajectories with more DOFs in the manipulator is mentioned in the Future Work paragraph, however it would be great if the authors could also include a very short description of potential ways they envision modifying or building on the method in order to expand to higher dimensional trajectories.