Scholars Spotlight

Please introduce yourself, school, country, grade, state, etc and your academic background. What  is your current school, and what drew you to the intersection of mechanical engineering and  control theory? Hello, my name is Tabina Hussain, I was born and raised in Washington State in the USA. I am  currently a Junior at Eastlake Highschool while simultaneously taking a part-time role in the running  start program at Central Washington University. Growing up in the Seattle region, Innovation and  creativity has always been around me flourishing and advancing. Being a witness to such innovation,  I naturally developed an effect to this and grew up to have a strong, passionate internal drive to  make an impact of my own. I developed the intention to innovate and create products to help my  community and give back to the place that nurtured my drive. That is when I developed a fondness  towards mechanical engineering, I was amazed at the industries’ creative freedom yet direct  impact on society. In mechanical engineering, I’ve always been fascinated by how things move and  how we can make machines behave intelligently. In high school, I’ve taken advanced math and  science courses including UW Precalculus, Honors Physics, and AP Physics 1 to help understand  this. Outside the classroom, I’ve been heavily involved in robotics. For the past 3 years, I’ve been participating in TSA, where I participated in engineering events. I also started the first F1 in Schools’ team at my old school Tesla Stem Highschool. During this program we learned how to manufacture  and engineer our own miniature cars through CAD. Currently, I am actively working to host Arduino  classes for kids aged 9-15 at my local community centers through my educational organization,  CircuitLab Arduino. What really drew me to the intersection of mechanical engineering and control  theory is the combination of building physical systems and making them perform precisely. On the  F1 in Schools team, I saw that even great mechanical designs can fail if the control system isn’t  good. I would notice that the robot would shake, overshoot, or become unstable. That frustration  turned into curiosity. I started learning about PID controllers, sensor fusion through Arduino, and  feedback systems, and I became uncaptured on how math and physics can be used to control real world motion. I thoroughly enjoy mechanical engineering as it is hands-on and creative and I am  encaptured by control theory as it gives me the intelligence that makes machines reliable and even  agile. This combination feels perfect in my goal for future work in robotics & autonomous vehicles,  which is why I’m very interested in pursuing mechanical engineering with a focus on dynamics and  control in college.

Describe your full experience at PREMIERE. With so many research opportunities available, why  did you choose to pursue a project on underactuated kinematics specifically through PREMIERE  Research Academy? During my time in the PREMIERE Research Academy, I gained comprehensive research experience  where I first learned about underactuated robots by reading numerous professional papers, then  developed my own model to solidify my understanding. Under the consistent guidance of my  mentor, I participated in weekly applied math tutoring sessions that helped me deeply understand  concepts such as eigenvectors, eigenvalues, derivations, and Lagrangian mechanics. I took parts of  my initial paper and transformed it into a new, more advanced research paper in which I expanded  the applied mathematics section, focusing on all the key factors that influence Underactuated  Kinematics and the Eigenvector Decomposition of a Self-Balancing Robotic Platform. I also created  my own detailed appendix to support the work. Throughout the process, I learned how to use  LaTeX while developing my initial paper, and near the end I mastered Jupyter Notebook through  the Anaconda Navigator platform to generate numerical simulations and seamlessly integrate  graphs into my final LaTeX document. I specifically chose this project on underactuated kinematics  through PREMIERE Research Academy because it perfectly aligned with my passion for the  intersection of mechanical engineering and control theory. While many research opportunities  exist, this one allowed me to dive deeply into a challenging and highly relevant area, underactuated  systems that appear in real-world applications like legged robots while receiving structured  mentorship and rigorous mathematical application training that most other programs do not offer  comprehensively at the high school level.

I would describe my overall experience at PREMIERE as rigorous yet extremely rewarding in  benefits. The content was initially very challenging, and the learning process was strenuous and  time-consuming, but every effort I put in helped me grow significantly, not only mathematically,  but also intellectually by teaching me to be more intentional and disciplined in my learning. For  future students, I strongly recommend being proactive and communicative with your mentor. If  you’re struggling with a concept, reach out early for additional support instead of waiting. I also  advise staying organized by tracking all assignments and due dates, then spacing out your work so  you can produce high-quality research rather than rushing at the last minute. Research requires  patience and attention to detail, especially when it comes to clearly communicating your findings  to others. Looking ahead, I plan to carry the knowledge, work ethic, and research skills I gained  from PREMIERE into college for the next steps of my educational path. I intend to major in  mechanical engineering, with a strong focus on animatronics and autonomous vehicles. Ultimately,  I hope to work with an innovative organization that has global impact and gives me the creative  freedom to design and develop advanced robotic systems.

What is your research topic about, how did you develop an interest in it, and in what ways has your  experience at PREMIERE Research Academy helped you achieve your research goals?  My research topic is “Underactuated Kinematics and Eigenvector Decomposition of a Self Balancing Robotic Platform.” In this project, I studied underactuated robotic systems which are  systems that have fewer actuators than degrees of freedom using a self-balancing two-wheeled  robot as the primary model. I focused on deriving the kinematic and dynamic equations using  Lagrangian mechanics, performed stability analysis through eigenvector and eigenvalue  decomposition, and examined the various physical and mathematical factors that influence the  behavior and control of these systems. I developed a strong interest in this topic through my hands on experience on my high school robotics team and personal projects. While building self-balancing  robots I repeatedly faced challenges with stability and control using limited arduino motors. This  frustration sparked my curiosity about how advanced robots like drones, legged robots, and  autonomous vehicles can achieve complex motions though being underactuated. Reading  professional research papers on the subject further deepened my fascination with difficult  mathematics behind controlling these systems. My time at the PREMIERE Research Academy was  essential and important in helping me achieve my research goals. The program provided me with  comprehensive mentorship and weekly applied math tutoring sessions that strengthened my  understanding of Lagrangian mechanics, eigenvalues, eigenvectors, and derivations. I started by  reading professional papers, then developed my own model to solidify my learning. I learned how  to use LaTeX for professional writing and Jupyter Notebook through Anaconda Navigator to run  numerical simulations and integrate graphs into my paper. Most importantly, I expanded my initial  paper into a much stronger final research document by significantly improving the applied  mathematics section and creating a detailed appendix. PREMIERE gave me the structure, skills, and  confidence to move from simply being interested in the topic to producing real, original research  work.

Your research departs from Newtonian methods in favor of the Euler–Lagrange formulation. Why  was the Lagrangian approach particularly suited for modeling a 2-DOF underactuated system? The Lagrangian approach is particularly well-suited for this 2-DOF underactuated cart-pendulum  system because it systematically derives the equations of motion from kinetic and potential  energies using generalized coordinates x, θ, automatically incorporating the holonomic constraints  and inertial coupling without explicitly resolving interaction forces at the pivot.

In your derivation of the pendulum’s velocity, you expanded the squared speed expression. Walk  us through how the coupling term involving cosine emerges mathematically, and explain how it physically represents the transfer of energy between the cart base and the pendulum body. The position is xₚ = x + L sin θ, yₚ = –L cos θ. Differentiating gives the velocity components: ˙xₚ = ˙x + L ˙θ cos θ, ˙yₚ = L ˙θ sin θ. The squared speed is then ˙xₚ² + ˙yₚ² = (˙x + L ˙θ cos θ)² + (L ˙θ sin θ)² = ˙x² + 2˙x L ˙θ cos θ + L²˙θ²  cos²θ + L²˙θ² sin²θ. Using the trigonometric identity cos²θ + sin²θ = 1, this simplifies to ˙x² + 2 L ˙x ˙θ cos θ + L² ˙θ². The cross term 2 L ˙x ˙θ cos θ (which becomes m L ˙x ˙θ cos θ in the total kinetic energy after  multiplying by m/2) is the coupling term. Physically, this term represents the instantaneous power exchange or energy transfer between the  carts translational motion and the pendulums rotational motion. When the base moves  horizontally (˙x) while the pendulum has angular velocity (˙θ), the relative velocity of the pendulum  mass has a component aligned with the base motion that depends on the instantaneous angle θ .  At θ ≈ 0 (upright), cos θ ≈ 1, so base acceleration strongly influences the pendulum’s tangential  speed, allowing kinetic energy to flow efficiently between the two bodies through inertial reaction  forces at the pivot. This coupling is what enables indirect control of the unactuated pendulum angle  thru base motion and is central to both the instability and the controllability of the self-balancing  platform.

Your paper utilizes the reduced state vector. Why was the horizontal position omitted from the  state-space model, and how did you justify the small-angle approximations used during the  derivation of the system matrix? The horizontal position was omitted from the reduced state vector because it behaves as a drift  coordinate that does not influence the local stability or instability dynamics near the upright  equilibrium. As explained in Section 4.1, absolute base position has no intrinsic mechanical need  for balance and the relevant quantities are base velocity, which determines support-point motion  relative to the center of mass, and the angular states. Including x would unnecessarily increase the  model order without adding information about the unstable manifold orstructure. The small-angle  were applied only during linearization which we can se in Section 4.2 to obtain the 3×3 model and  system matrix A. They are standard and justified for local analysis around the equilibrium θ = 0,  where accurately capturinh the first-order behavior having the onset of instability. The full  nonlinear equations were derived exactly from the Lagrangian, preserving geometric until the  point where linear modal analysis becomes appropriate. This makes the clean eigenvalues and  eigenvector while remaining a valid local approximation, as later confirmed by numerical  comparisons between linear and nonlinear simulations.

You argue that the unstable eigenvector defines the kinematic direction of collapse. How did you derive the proportional relationship between the base velocity and the pendulum pitch angle θ during the instability phase? The proportional relationship ˙x / θ = −(m g) / (γ M) during the initial (linear) stage of falling is derived directly by computing the unstable eigenvector v₃ associated with the positive eigenvalue λ = +γ of the reduced system matrix A. This means that, in the early phase of collapse, a positive pitch angle θ is accompanied by a specific negative base velocity x. The relationship is purely kinematic and geometric and any stabilizing controller must overcome this natural direction of motion. This is exactly what Section 6.3 calls the “kinematic direction of collapse.”

One of the most valuable pieces of technical feedback I received from my mentor concerned the  derivation of the Euler–Lagrange equations in Section 3.3. they specifically flagged that section  with the comment: “Euler–Lagrange Equations (Needs Improvement) — This is the most important  area to fix. You did not fully show the intermediate steps required... In particular, the time  derivatives (e.g., d/dt(mL˙θ cos θ)) were not expanded step-by-step using the product rule, and  some cancellations appear without being explicitly demonstrated.” This feedback was extremely  helpful. Following their advice, I significantly expanded that subsection by carefully applying the  product rule to each term, demonstrating every cancellation, and clarifying the origin of the −mL  ˙θ² sin θ term. These additions made the symbolic derivations far more rigorous and transparent,  greatly improving the quality of the paper and satisfying the assignment’s requirements for  detailed step-by-step reasoning. This revision also strengthened the logical flow into the  linearization and eigenvector analysis that followed.

Working through the full Lagrangian derivation, linearization, and especially the symbolic  computation of the unstable eigenvector gave me deep knowledge for how geometric wualities from linear algebra can reveal the fundamental “direction of collapse” in underactuated systems.  This experience solidified my desire to pursue a rigorous undergraduate program in Robotics,  Mechanical Engineering, or Electrical Engineering with a strong control theory focus. I now actively  seek programs that offer advanced coursework in nonlinear control, hybrid systems, and  optimization-based control. Long-term, I aspire to become a researcher and engineer who designs  robust, energy-efficient underactuated robots that operate reliably in real-world environments.  This project taught me that true completionlies not just in stabilizing systems, but in understanding  their intrinsic geometry and instabilities. I want to contribute to next-generation control framework, and help advance safe and capable autonomous machines. This research experience has  transformed my ambition from simply studying robotics to actively pushing its limits.

My top advice for current and future PREMIERE Research Academy students is to embrace rigor  and clarity as non-negotiable standards from the very beginning. When I submitted my early drafts,  my mentor gave me critical feedback on the Euler–Lagrange derivations, pointing out that I had  skipped important intermediate steps like fully expanding the product rule when taking time  derivatives. That feedback, though tough, became one of the most valuable parts of the entire  process. It forced me to slow down and show every cancellation explicitly, and truly understand  why each term appeared. The result was not only a stronger paper but also a much deeper personal  mastery of the material. So here is my practical advice: • Be obsessive about showing your work • Seek and genuinely listen to critical feedback • Connect theory to intuition and simulation. Don’t stop at the equations. • Choose a topic you’re genuinely excited about. The late nights and challenging math  become much more bearable when you like the system you’re studying. • Finally, remember that this academy is training you not just to complete a project, but to  think and communicate like a real researcher